On Uniformly Random Discrete Interlacing Systems
Asymptotics and Universal Edge Fluctuations with Applications to Lozenge Tiling
Models
ERIK DUSE
Doctoral Thesis
Stockholm, Sweden 2015
TRITA-MAT-A 20015:12
ISRN KTH/MAT/A–15/12-SE
ISBN 978-91-7595-732-6
KTH
Institutionen för Matematik
100 44 Stockholm
SWEDEN
Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges
till offentlig granskning för avläggande av teknologie doktorsexamen i matematik
veckodag den 4 december 2015 kl 13.00 i D3, Kungl Tekniska högskolan, Lindstedtsvägen 5, Stockholm.
c Erik Duse, 2015
Tryck: Universitetsservice US AB
iii
Abstract
This thesis concerns uniformly random discrete interlacing particle systems and their connections to certain random lozenge tiling models. In particular it contains the first derivation of a relatively unknown universal scaling
limit, which we call the Cusp-Airy process, of certain lozenge tiling models
at a cusp point. In addition it contains a characterization of the geometry
of the macroscopic behavior of uniformly random discrete interlaced particle systems that, although not complete, shows many new and interesting
features.
iv
Sammanfattning
Denna avhandling behandlar likformigt slumpmässiga system av sammanflätade partiklar (eng: interlacing particles) och deras koppling till vissa rombiska tesselerings modeller. Den innehåller i synnerhet den första härledningen av en relativt okänd universell skalningsgräns, som vi kallar Cusp-Airy
processen, för vissa rombiska tesselerings modeller vid en spetspunkt. Den
innehåller dessutom en karakterisering av geometrin för det makroskopiska
beteendet hos likformigt slumpmässiga system av sammanflätade partiklar,
som trots att den inte är fullständig, visar på nya och intressanta egenskaper.
Contents
Contents
v
Acknowledgements
vii
Part I: Introduction and Summary
1 Introduction
1.1 Random Lozenge Tilings . . . . . . . . . . . . . . . . . . . .
1.2 Cut-Corner Hexagon Dimer Model and Related Models . .
1.3 Discrete Interlacing Systems . . . . . . . . . . . . . . . . . .
1.4 Discrete Orthogonal Polynomial Ensembles and Equilibrium
1.5 Relation To Continuous Interlacing Model . . . . . . . . . .
2 Summary
2.1 Paper
2.2 Paper
2.3 Paper
of
A
B
C
2
. . . . .
2
. . . . .
4
. . . . . 10
Measures 38
. . . . . 49
Results
53
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
References
55
v
Acknowledgements
First and foremost I would like to express my gratitude to my scientific advisor
Kurt Johansson for all his support and encouragement during these past years. In
particular I thank him for his generosity in sharing his mathematical knowledge
and his friendship. I would also like to thank Anthony Metcalfe for a fruitful
cooperation. In addition I would like to thank Maurice Duits and Sunil Chhita for
many useful and interesting discussions regarding my research. I am also grateful
to Professor Richard Kenyon at Brown University for letting me use his figure of a
simulation of a random tiling of the cut-corner hexagon, figure 1 in the introduction.
My graduate studies were funded by the grant KAW 2010.0063 from the Knut
and Alice Wallenberg Foundation. For this I am very grateful.
I would like to thank all the friends I have gained at the department of mathematics at KTH during my years as a graduate student. In particular, I would like
to thank Martin Strömqvist, Joel Andersson, Andreas Minne, Gaultier Lambert,
Mauriusz Hynek, Christopher Svedberg, Oscar Forsman, Katharina Heinrich, Sebastian Öberg, Samuel Holmin, André Laestedius, Gustav Sændén Ståhl and Antti
Haimi for all interesting discussions over the years.
Last but not least, I would like to thank my family for all support during these
years. In particular I would like to express my gratitude and love to my fiancée
Frida, my parents Alfred and Maria and my grandmother Margareta. This thesis
would not have been made possible without you.
Finally I would like to dedicate this thesis to my late grandfather Erik Öhlin,
who served as a great inspiration to me during my childhood.
vii
1
Introduction
1
2
1.1
Random Lozenge Tilings
Figure 1.1: A simulation of a random lozenge tiling of the cut-corner hexagon
model.
3
The Cut-Corner Hexagon Dimer Model
This thesis was born out of an attempt to understand a special feature of the cutcorner hexagon dimer model. More precisely, consider a regular hexagon with unit
side length, and remove from one corner a rhombi of side length 1{2. We call this
polygon the cut-corner hexagon. See figure 1.2 below.
Figure 1.2: The cut-corner regular hexagon.
Consider tiling the cut-corner hexagon by the rhombi forming the faces of a
cube under the projection in the p1, 1, 1q direction with side lengths 1{p2nq. The
rhombi, or lozenges, are shown in figure 1.3.
Y
B
R
Figure 1.3: Three types of lozenges with sides of length 1.
Now, consider all possible such tessellations. Put the uniform probability distribution on the set of all tessellations. One may now ask many interesting questions
about this model. For example, one may ask what a “typical” tessellation look like
when n is very large, if there is any such thing. Indeed, a simulation of a typical
tessellation is shown in figure 1 in [KO07] and reprinted in this introduction as
figure 1.1 by permission of Professor Richard Kenyon. From, this figure one may
immediately observe some special features. To start with one observes a brick-like
pattern of only one type of lozenge in each of the corners. These frozen regions
are interrupted by a curve. Inside the region of this curve one observes all types
of lozenges. One also observes that the curve touches the sides of the polygon tangentially, see figure 1.1. By letting n become large, this curve will look more and
4
more like a smooth curve. In particular, this smooth curve will look very similar to
a cardioid curve, again see figure 1.1. Moreover, this curve has a singularity in the
form of a cusp. The starting point of this thesis was to attempt to understand the
local microscopic behavior of the stochastic process of the lozenges in the vicinity
of the cusp as n Ñ 8. Before discussing how this question was resolved, we will in
the next sections first describe more general lozenge tiling models and their relation
to interlaced particle systems.
1.2
Cut-Corner Hexagon Dimer Model and Related Models
Decomposition of Polygons into Interlacing Systems
Consider first a tessellation of a regular hexagon. An example of such is shown
in figure 1.4. In figure 1.5 we also depict the “frozen regions” and the “frozen
boundary” separating the frozen region from from the “liquid region”. Also, the
asymptotic shape of the frozen boundary is shown.
1
1
1
1
1
Y
B
R
1
Figure 1.4: Left: A regular hexagon with sides of length 1.
Middle: The three different types of lozenges with sides of length 1{n.
Right: An example tiling when n “ 4.
5
1
R
B
Y
1
1
1
1
Y
B
R
1
Figure 1.5: Left: The frozen boundary of the example tiling of figure 1.4.
Right: The asymptotic shape of the frozen boundary of a “typical random tiling”
as n Ñ 8.
We will now restrict our attention to the configuration of yellow tiles. We
see that we may encode the configuration of yellow tiles as interlacing systems of
particles. In in figure 1.6 this is done in two different ways. In the figure to the
left in figure 1.6, we encode the configuration of yellow tiles as two interlacing
systems. One interlacing system between row 1 and row 4, and one interlacing
system between row 7 and row 4, being glued together along their common row,
row 4. Moreover, since we pick tessellations of the regular hexagon uniformly at
random, the configuration of tiles at row 4 is a random configuration. On the other
hand, in the figure to the right in figure 1.6 we encode the configuration of yellow
tiles as one interlacing system by adding virtual particles on the side of the polygon.
Now, the configuration of yellow tiles at row 8, the “top line”, is deterministic as
opposed to the configuration of tiles on row 4.
We may now note that the configuration of yellow tiles entirely fixes the configurations of the other tiles, the red and the blue tiles. This can be seen as follows.
In between every row of yellow tiles we may we have a row of red and blue tiles.
Firstly, we see that the position of the yellow tile on the first row fixes the positions
of the tiles on first row of red and blue tiles. Secondly, we see that the position of
yellow tiles on the first and the second row fixes the position of red and blue tiles
on the the second row of blue and red tiles. The process may be continued until we
reach the “top row” of the first interlacing system, row 4, and the position of the
tiles on row 3 and row 4 have determined the position of the red and blue tiles on
the fourth row of the rows of red and blue tiles. We have now fixed the position of
all the tiles in the lower interlacing system. We may now repeat the process in the
upper interlacing system, using that the position of the yellow tile on row 7 fixes
position of red and blue tiles on row 8 and so on. This completes the tessellation
of the regular hexagon, given the position of the yellow tiles. In fact we notice that
the choice of considering yellow tiles was arbitrary, we may in fact equally well have
6
chosen to consider the configurations of red or the blue tiles instead.
row 8
row 7
row 6
row 5
row 4
row 3
row 2
row 1
Figure 1.6: Left: Equivalent interlaced particle configuration of the example tiling
of figure 1.4.
Right: Equivalent interlaced particle configuration with added deterministic
lozenges/particles. The unfilled circles represent the deterministic particles.
We will now be more precise on how we encode the positions of the yellow tiles
prq
as an interlaced particle system. Let yi denote the position of the i:th particle
pnq
pnq
on the r:th row and let βi :“ yi denote the position of the particles on the top
line, indicated by unfilled circles inside the tiles. Then the particles on row r ` 1
will interlace with the particles on row r according to
pr`1q
y1
prq
pr`1q
ą y1 ą y2
prq
pr`1q
ą y2 ... ą yrprq ą yr`1 ,
for every r “ 1, ..., n ´ 1.
We may now try to repeat the idea of encoding lozenge tessellations of the
regular cut-corner hexagon by interlacing systems. This can be done in the three
different ways depicted in figure (1.7).
7
Figure 1.7: The picture to the left depicts a decomposition of the cut-corner hexagon
into two interlacing regions of yellow tiles. The picture in the middle depicts a
decomposition of the cut-corner hexagon into two interlacing regions of red tiles.
The picture to the right depicts a decomposition of the cut-corner hexagon into two
interlacing regions of blue tiles
In particular, the methods developed in this thesis will apply to those polygons
that can be decomposed into two interlacing regions for at least one of the three
different types of tiles. For a more general decomposition of such a polygon see
figure 1.10 in the next section.
It will be convenient when considering interlacing system to make a coordinate
transformation according to figure 1.8. For more details see section 1.4 in [DM15a]
and section 2.1 in [Pet14] .
Figure 1.8: Coordinate transformation of lozenge tiles.
After the coordinate transformation, figure 1.4 becomes figure 1.9. Furthermore
the interlacing condition between row r ` 1 and row r has changed into
pr`1q
y1
prq
pr`1q
ě y1 ą y2
prq
pr`1q
ě y2 ... ě yrprq ą yr`1 .
8
Figure 1.9: An example tiling and its equivalent interlaced particle configuration
after the coordinate transformation. The unfilled circles represent the deterministic
lozenges/particles.
Discrete Orthogonal Polynomial Ensembles
As was described in the previous section we are interested in those tiling models
that can be decomposed into two regions, such that after possibly adding virtual
particles, these regions become interlacing regions of the type describe in section
1.2, glued together along a common line as depicted in figure 1.10.
T1
Interlacing direction
T2
Figure 1.10: Decomposition of a polygon into two interlacing regions T1 and T2
glued together along the thick black line. The blue dots indicate the positions of
virtual particles/tiles and the black dots indicate the positions of ordinary particles/tiles.
Recall that the number of interlacing configurations with a given top line con-
9
figuration py1 , y2 , ...., yN q P ZN is given by Weyl’s dimension formula in [Wey39] for
the irreducible characters of the unitary group U pnq,
ś
ź
1
1ďiăjďN |yi ´ yj |
7
:“
N py1 , ..., yN q “ ś
|yi ´ yj |.
(1.2.1)
2CN 1ďi,jďN
1ďiăjďN |i ´ j|
i‰j
Let py1 , y2 , ...., yN q be the positions of the particles/tiles on the intersecting thick
black line as in figure 1.10. Let V be the index set for the virtual or frozen particles
and let F be the index set for the free particles, so that |V| ` |F| “ N and yi
is a virtual particle if i P V and free otherwise. Assume that |F| “ n, and let
g : t1, ...., nu Ñ F be a set bijection such that xi :“ ygpiq , and x1 ă x2 ă ... ă xn .
Furthermore, the virtual particles are densely packed, which implies that they vill
form wedge shaped frozen regions. However, the fact that the two interlacing regions
T1 and T2 need not be symmetrical implies that we need not have frozen regions
on both sides of the intersecting black line. Let VL Ď V be the index set of those
virtual particles such that they form a frozen region to the left, and let VR Ď V be
the index set of those virtual particles such that they form a frozen region to the
right. Then by (1.2.1), the number of interlacing configuration with a given fixed
configuration of free particles at positions px1 , ..., xn q is given by
ź
1
N 7 px1 , ..., xn q “
|yi ´ yj |
2CN 1ďiăjďN
ź
ź
ź
ź
1 ź
“
|yi ´ yj |2
|yi ´ yj |
|yi ´ yj |
|yi ´ yj |
|yi ´ yj |
2CN i,jPF
iPF
iPF
i,jPV
i,jPV
i‰j
“
1
2CN
ź
1ďk,lďn
k‰l
jPVL
|xk ´ xl |2
L
jPVR
n ź
ź
|xk ´ yj |
k“1 jPVL
R
i‰j
n ź
ź
i‰j
|xk ´ yj |
k“1 jPVR
ź
|yi ´ yj |
i,jPVL
i‰j
ź
|yi ´ yj |.
i,jPVR
i‰j
Now, consider the set of all possible lozenge tessellations of the the original polygon.
One easily sees that each such tessellation is in a bijective correspondence with two
interlacing configurations on T1 and T2 with the same configuration of free particles
xpnq “ px1 , x2 , ..., xn q on their common top line. In particular, xi P nΣ X Z for each
i “ 1, ..., n, where Σ is a finite union of intervals defined by the tiling model. Let
Cn denote the set of all configurations of free particles. Consider the set of all
lozenge tessellations of the polygon with uniform distribution. Then this induces a
probability distribution on Cn , given by
ppnq rpx1 , ...., xn qs “ Prparticles at positions x1 , x2 , ..., xn s “
N 7 px1 , ..., xn q
ÿ
.
N 7 px1 , ..., xn q
px1 ,...,xn qPCn
Let
wn pxq :“
ź
jPVL
|x ´ yj |
ź
jPVR
|x ´ yj |.
10
Then
ppnq rpx1 , ...., xn qs “
1
Zn
ź
pxi ´ xj q2
1ďiăjďn
n
ź
wn pxi q,
(1.2.2)
i“1
ř
ś
śn
where Zn “ px1 ,...,xn qPCn 1ďiăjďn pxi ´xj q2 i“1 wn pxi q. Associated with a particular weight function wn pxq is a class of discrete orthogonal polynomials tpn,k pxquk
defined according to
ÿ
pn,k pxqpn,l pxqwn pxq “ δkl .
(1.2.3)
xi PnΣXZ
Such particle processes are called discrete orthogonal polynomial ensembles, DOPE,
and have been studied e.g. inř[BKMM07]. In particular if one consider the random
n
empirical measure µn “ n1 i“1 δxi {n , then µn á µλV in probability, where the
measure µλV P Mλ1 pΣq is the unique solution of the constrained variational problem
"ˆ
*
ˆ
´1
min tIV rνsu “ min
log |x ´ y| dνpxqdνpyq `
V pxqdνpxq ,
νPMλ
1 pΣq
νPMλ
1 pΣq
ΣˆΣ
Σ
(1.2.4)
where V pxq “ limnÑ8 ´n´1 logpwn pxqq. Here, Mλ1 pΣq is the set of positive Borel
measures ν, such that supppνq Ă Σ, }ν} “ 1 and ν ď λ, where λ is the Lebesgue
measure. The constrained variational problem is discussed in more detail in section
1.4.
Again we emphasize that in these models we have a random empirical top line
measure. We will not discuss the issues arising from this fact further in introduction, but refer the interested reader to [DJM15]. For more on discrete orthogonal
polynomial ensembles see [BKMM07] and [Fer].
1.3
Discrete Interlacing Systems
Discrete Interlacing Systems and Determinantal Point Processes
We saw in the previous section that the study of certain random lozenge tiling
models with uniform probability could be reduced to the study of certain discrete
interlacing models. In this section we therefore give a more careful introduction to
these systems. A discrete Gelfand-Tsetlin pattern of depth n is an n-tuple, denoted
py p1q , y p2q , . . . , y pnq q P Z ˆ Z2 ˆ ¨ ¨ ¨ ˆ Zn , which satisfies the interlacing constraint
pr`1q
y1
prq
ě y1
pr`1q
ą y2
prq
ě y2
pr`1q
ą ¨ ¨ ¨ ě yrprq ą yr`1 ,
denoted y pr`1q ą y prq , for all r P t1, . . . , n ´ 1u. For each n ě 1, fix xpnq P Zn with
pnq
pnq
pnq
pnq
pnq
pnq
pnq
x1 ą x2 ą ¨ ¨ ¨ ą xn and an “ xn and bn “ x1 , and consider the following
11
probability measure on the set of patterns of depth n:
"
1
1 ; when xpnq “ y pnq ą y pn´1q ą ¨ ¨ ¨ ą y p1q ,
p1q
pnq
νn rpy , . . . , y qs :“
¨
0 ; otherwise,
Zn
where Zn ą 0 is a normalisation constant. This can equivalently be considered as
a measure on configurations of interlaced particles in Z ˆ t1, . . . , nu by placing a
particle at position pu, rq P Zˆt1, . . . , nu whenever u is an element of y prq . Thus, νn
is the uniform probability measure on the set of all such interlaced configurations
with the particles on the top row in the deterministic positions defined by xpnq .
Also note that due to the interlacing constraint, the interlacing particle system is
contained inside the polygon nPn , where
pnq
Pn “ tpχ, ηq P R2 : 0 ď η ď 1, χ ` η ´ 1 ě apnq
n , χ ď bn u
(1.3.1)
We now construct a related probability space, the determinantal structure of
which is more convenient to examine. Consider all tuples, pz p1q , . . . , z pn´1q q P Zn ˆ
¨ ¨ ¨ ˆ Zn with
pr`1q
z1
prq
ě z1
pr`1q
ą z2
prq
ě z2
ą ¨ ¨ ¨ ą znpr`1q ě znprq ,
for all r, also denoted z pr`1q ą z prq . Fix z p0q :“ pxn ` n ´ 1, . . . , xn ` 1, xn q and
define the following probability measure on the set of all such pn ´ 1q-tuples:
"
1
1 ; when xpnq ą z pn´1q ą ¨ ¨ ¨ ą z p1q ą z p0q ,
1
p1q
pn´1q
νn rpz , . . . , z
qs :“ 1 ¨
0 ; otherwise,
Zn
(1.3.2)
where Zn1 ą 0 is a normalisation constant.
Consider the relationship between the spaces. First note that, whenever xpnq “
y pnq ą y pn´1q ą ¨ ¨ ¨ ą y p1q for some py p1q , y p2q , . . . , y pnq q P Z ˆ Z2 ˆ ¨ ¨ ¨ ˆ Zn , then
prq
x1 ě y1 ą ¨ ¨ ¨ ą yrprq ą xn ` n ´ r ´ 1,
for all r ď n. Whenever x ą z pn´1q ą ¨ ¨ ¨ ą z p1q ą z p0q for some pz p1q , . . . , z pn´1q q P
Zn ˆ Zn ˆ ¨ ¨ ¨ ˆ Zn ,
=
=
prq
ą zr`2
xn `n´r´1
prq
xn `n´r´2
ą ¨ ¨ ¨ ą znprq
...
(1.3.3)
=
prq
prq
x1 ě z1 ą ¨ ¨ ¨ ą zrprq ą zr`1
xn
prq
for all r ď n ´ 1. We refer to z1 , . . . , zr as the free particles of z prq , and
prq
prq
zr`1 , . . . , zn as the deterministic particles. Note the natural bijection between
p1q
tpy , . . . , y pnq q P Z ˆ Z2 ˆ ¨ ¨ ¨ ˆ Zn : x “ y pnq ą y pn´1q ą ¨ ¨ ¨ ą y p1q u and
tpz p1q , . . . , z pn´1q q P Zn ˆ Zn ˆ ¨ ¨ ¨ ˆ Zn : x ą z pn´1q ą ¨ ¨ ¨ ą z p1q ą z p0q u: remove y pnq from each n-tuple py p1q , . . . , y pnq q and map the remaining components,
prq
prq
y prq “ py1 , . . . , yr q for each r ď n ´ 1, individually as,
prq
y prq ÞÑ py1 , . . . , yrprq , xn ` n ´ r ´ 1, xn ` n ´ r ´ 2, . . . , xn q.
12
The measure νn1 is induced by the measure νn under this bijective map. The probabilistic structure of particles in the first space (measure νn ) is therefore identical
to the probabilistic structure of the free particles in the second space (measure νn1 ).
From now on we restrict to the second space.
A more convenient expression for νn1 can be obtained from the work of Warren,
[War07]:
"
”
ın
1 ; when z pr`1q ą z prq ,
“
det 1zpr`1q ězprq
0 ; otherwise,
j
i
i,j“1
for all r. Equation (1.3.2) thus gives,
νn1 rpz p1q , . . . , z pn´1q qs “
n´1
”
ın
1 ź
prq pr`1q
det
φ
pz
,
z
q
,
r,r`1
i
j
Zn1 r“0
i,j“1
(1.3.4)
where z pnq :“ x, and
φr,r`1 pu, vq :“ 1věu ,
for all r and u, v P Z. We may define coupling matrices Ar,r`1 “ Ar,r`1 pz prq , z pr`1q q,
connecting configurations z prq on row r with configurations z pr`1q on row r ` 1 by
letting
prq
pr`1q
pAr,r`1 pz prq , z pr`1q qqij :“ φr,r`1 pzi , zj
q.
Using the coupling matrices we see that (1.3.4) can be written as
νn1 rpz p1q , . . . , z pn´1q qs

„ n´1
ź
1
prq pr`1q
“ 1 det
Ar,r`1 pz , z
q .
Zn
r“0
(1.3.5)
Note, each pz p1q , . . . , z pn´1q q P Zn ˆ Zn ˆ ¨ ¨ ¨ ˆ Zn can be equivalently considered
as a configuration of particles in Z ˆ t1, . . . , n ´ 1u by placing a particle at position
pu, rq P Z ˆ t1, . . . , n ´ 1u whenever u is an element of z prq . The measure νn1
in equation (1.3.5) therefore defines a random point process on configurations of
particles in Z ˆ t1, . . . , n ´ 1u. The Eynard-Mehta theorem, see Proposition 2.13
of Johansson, [Joh06], proves that this process is determinantal with correlation
kernel,
Kn ppu, rq, pv, sqq “ K̃n ppu, rq, pv, sqq ´ φr,s pu, vq,
(1.3.6)
for all r, s P t1, . . . , n ´ 1u and u, v P Z, where
K̃n ppu, rq, pv, sqq :“
n
ÿ
k,l“1
and
φr,s pu, vq :“ 0 when s ď r,
pnq
p0q
φr,n pu, zk qpA´1 qkl φ0,s pzl , vq,
13
φr,s pu, vq :“ 1věu when s “ r ` 1,
ÿ
φr,r`1 pu, z1 qφr`1,r`2 pz1 , z2 q ¨ ¨ ¨ φs´1,s pzs´r´1 , vq when s ą r ` 1,
φr,s pu, vq :“
z1 ,...,zs´r´1
APC
nˆn
p0q
pnq
with Akl :“ φ0,n pzk , zl q for all k, l.
Note that, for all r, s P t1, . . . , nu and u, v P Z, one may in fact compute φr,s pu, vq
to get
φr,s pu, vq “ 1sąr hv´u pp1qs´r q,
for all r, s P t1, . . . , nu and u, v P Z, and where hk denotes the homogeneous symmetric polynomials and where by convention hk :“ 0 whenever k ă 0. Using the
special form of φr,s pu, vq, we where in [DM15a] able to compute K̃n ppu, rq, pv, sqq,
and to derive an alternative form for φr,s pu, vq. We got
śu´1
n
v
ź ˆ l ´ xi ˙
pn ´ sq! ÿ ÿ
j“u`r´n`1 pxk ´ jq
K̃n ppu, rq, pv, sqq “
1xk ěu śv
,
pn ´ r ´ 1q! k“1 l“v´n`s
j“v´n`s,j‰l pl ´ jq i‰k xk ´ xi
(1.3.7)
and
śu´1
n
v
ź ˆ l ´ xi ˙
pn ´ sq! ÿ ÿ
j“u`r´n`1 pxk ´ jq
ś
.
φr,s pu, vq “ 1věu
pn ´ r ´ 1q! k“1 l“v´n`s vj“v´n`s,j‰l pl ´ jq i‰k xk ´ xi
It is interesting and surprising to note that both K̃n ppu, rq, pv, sqq and φr,s pu, vq
have the same form. Also note that the fixed top row and the interlacing constraint
implies that it is sufficient to restrict to those positions, pu, rq, pv, sq P Zˆt1, . . . , n´
pnq
pnq
1u, with u ě xn ` n ´ v and v ě xn ` n ´ s. We finally note that in terms of
lozenge tilings as depicted in figure 1.9, the correlation kernel Kn is the correlation
kernel for the positions of the yellow tiles. When we want to emphasize this fact
we will sometimes write KnY . In [DJM15] we showed how to get correlation kernels
of the other tiles through the particle transformations
KnR ppu, rq, pv, sqq “ ´KnY ppu, rq, pv, s ´ 1qq
(1.3.8)
KnB ppu, rq, pv, sqq
(1.3.9)
“
KnY ppu, rq, pv
` 1, s ´ 1qq.
Here, KnR is a correlation kernel for the red tiles and KnB is a correlation kernel for
the blue tiles.
At this point it is advantageous to pause for a moment and reflect upon the fact
why we where able to compute the correlation kernel Kn ppu, rq, pv, sqq. We see that
it depended on two critical steps:
• We were able to compute the n-fold matrix product A and get ‘nice’ formulas
for the matrix elements pAqij .
14
• We were able to compute the inverse matrix A´1 , which by Cramer’s rule
reduces to computing determinants. The fact that we could compute the
determinants in turn depended critically on the form of the matrix elements
Aij .
For more complicated determinantal point processes, neither the first nor the
second step is usually feasible. One could ask why the uniformly random discrete
interlacing model has this integrable structure. We will not give a precise answer
to this question in the thesis but we note that interlaced particle systems fit within
the broader framework of Schur process. See for example [OR03] and [Bor11]. We
also note that there is a close connection between random interlaced particles and
Markov chains, see [BF15].
Asymptotic Limit Shapes
What do we mean by asymptotic limit shapes of discrete interlaced particle systems? In analogy with statistical physics we may view the asymptotic limit shapes
as the boundaries of different types of “phases”, i.e., microscopic behavior of the
underlying systems as n Ñ 8. The phases that can appear are called liquid or solid
state. Of course the limit n Ñ 8 is an idealization of a true system where n is very
large. Therefore, the asymptotic limit shape only exists in an idealized sense.
For many models from statistical physics it turns out that when one consider
various scaling limits of the microscopic system, i.e. one zoom in on the stochastic
system at a certain scale, one gets universal scaling limits. By universal scaling
limits we mean that the limit does not depend on the particular details of the
original statistical model and that the same limits occur in many different models.
For a physically realistic system the limit n Ñ 8 of the original microscopic system
can then be approximated by the universal scaling limit in much the same way as
the the sum of n identically distributed independent random variables, rescaled by
one over the square root of n, can be approximated by the normal distribution.
Which universal scaling limit one gets will depend on where one is looking in the
system. Since these limits can change in a discontinuous way depending on where
one is looking one expects to see different phases. In particular one expects that
for n large enough the microscopic behavior of system changes in an abrupt way as
one crosses the asymptotic boundary between different types of phases.
Now how is the asymptotic limit shape of uniformly discrete interlaced patterns
determined? In what follows we will change notation somewhat. Change xpnq “
px1 , ...., xn q Ñ β pnq “ pβ1 , ..., βn q and ppu, rq, pv, sqq Ñ ppx1 , y1 q, px2 , y2 qq. Since
this system is determinantal, the correlation functions are given by
ρppx1 , y1 q, px2 , y2 q, ..., pxm , ym qq :“ Prparticles at positions px1 , y1 q, px2 , y2 q, ..., pxm , ym qs
“ detrKn ppxi , yi q, pxj , yj qqsm
i,j“1 .
(1.3.10)
Therefore, various scaling limits of (1.3.10) (with m fixed) are entirely determined
by the correlation kernel Kn ppxi , yi q, pxj , yj qq. To analyze the scaling limits, in
15
[DJM15], we gave an integral representation (1.3.11) below of the correlation kernel
Kn based on the formula (1.3.7). This reduces the problem of studying limits of
the kernel (1.3.7) to the problem of determining the asymptotics of double contour
integrals. By the method of steepest descent in complex analysis, this problem in
1
1
turn reduces to determining the roots of certain functions fn,1
pwq, fn,2
pzq. Under
certain mild assumptions the function fn,i for i “ 1, 2 converges to an asymptotic
function f . Therefore the problem of determining the limit shapes of discrete
interlacing systems can be put in a bijective correspondence to a certain asymptotic
function f . This function will of course depend on the coordinates of the position
of where one is taking this limit and the weak limit of the sequence of point masses
n
1 ÿ
δ pnq to be defined below. The double contour integral representation of Kn
n i“1 βi
given in [DJM15] is
Kn ppx1 , y1 q, px2 , y2 qq “
˛
˛
śx2 ´1
n ˆ
pnq ˙
1 ź w ´ βi
k“x `y ´n`1 pz ´ kq
dw śx1 2 2
pnq
Γn
γn
z ´ βi
k“x1 `y1 ´n pw ´ kq w ´ z i“1
śx2 ´1
˛
˛
n ˆ
pnq ˙
pn ´ y1 q!
1
1 ź w ´ βi
k“x2 `y2 ´n`1 pz ´ kq
´ 1x1 ěx2
dz
dw śx1
,
pnq
pn ´ y2 ´ 1q! p2πiq2 Γ1n
γn
z ´ βi
k“x1 `y1 ´n pw ´ kq w ´ z i“1
(1.3.11)
1
pn ´ y1 q!
1x1 ăx2
pn ´ y2 ´ 1q! p2πiq2
dz
pnq
pnq
ě x2 u but
pnq
1
not the set tβj ď x2 ´1u, and Γn is a counterclockwise oriented contour containing
pnq
pnq
pnq
tβj : βj ă x2 u but not the set tβj ě x2 ` 1u. In addition, γn contains the set
tx1 ` y1 ´ n, ..., x1 u and Γn or Γ1n . The integration contours are shown in figure
where Γn is a counterclockwise oriented contour containing tβj
: βj
1.11.
γn
Γ1n
an
Γn
x2
bn
pnq
Figure 1.11: Integration contours. Here, an “ βn
pnq
and β1
“ bn .
One should note that the correlation kernel of a determinantal point process is
not uniquely defined. Indeed, if Kn ppx1 , y1 q, px2 , y2 qq is a correlation kernel of a de-
16
terminantal point process, so is Kn1 ppx1 , y1 q, px2 , y2 qq “
wpx1 , y1 q
Kn ppx1 , y1 q, px2 , y2 qq,
wpx2 , y2 q
since
m
wpxi , yi q
Kn ppxi , yi q, pxj , yj qq
“ det
wpxj , yj q
i,j
śm
i“1 wpxi , yi q
detrKn ppxi , yi q, pxj , yj qqsm
“ śm
i,j
wpx
,
y
q
j j
i“1
„
detrKn1 ppxi , yi q, pxj , yj qqsm
i,j
“ detrKn ppxi , yi q, pxj , yj qqsm
i,j .
Let
pnq
JΓn γn ppx1 , y1 q, px2 , y2 qq “
1
p2πiq2
˛
˛
dz
1Γ
n n
śx2 ´1
ˆ
˙
n
pnq
1 ź w ´ βi {n
k“x `y ´n`1 pz ´ k{nq
dw śx1 2 2
pnq
1γ
z ´ βi {n
k“x1 `y1 ´n pw ´ k{nq w ´ z i“1
n n
(1.3.12)
for all n P N. The integrand can be written as
exppnfn,1 pw; x1 , y1 q ´ nfn,2 pz; x2 , y2 qq
,
w´z
(1.3.13)
for all w, z P CzR, where
ˆ
˙
ˆ
˙
x1
n
ÿ
βi
1
j
1 ÿ
log w ´
´
log w ´
,
n i“1
n
n j“x `y ´n
n
1
1
ˆ
˙
˙
ˆ
xÿ
n
2 ´1
1 ÿ
βi
1
j
fn,2 pz; x2 , y2 q :“
log z ´
´
,
log z ´
n i“1
n
n j“x `y ´n`1
n
fn,1 pw; x1 , y1 q :“
2
2
and log denotes the principal logarithm. Let
µn “
n
1 ÿ
δ pnq
n i“1 βi {n
be the empirical measure associated with the distribution of particles on the top
line y pnq “ β pnq . Furthermore, let
λn “
1 ÿ
δ .
n kPZ k{n
We can then write
ˆ
fn,1 pw; x1 , y1 q “
ˆ
x1 {n
logpw ´ xqdµn pxq ´
ˆ
fn,2 pw; x2 , y2 q “
logpw ´ xqdλn pxq
x1 {n`y1 {n´1
R
ˆ
x2 {n´1{n
logpw ´ xqdµn pxq ´
R
logpw ´ xqdλn pxq.
x2 {n`y2 {n´1`1{n
17
Assume that for i “ 1, 2 we have limnÑ8 n1 pxi , yi q “ pχ, ηq and that µn converges weakly to a Borel measure µ, such that µ has compact support. Then for
i “ 1, 2 we have
ˆ
ˆ χ
lim fn,i pw; xi , yi q :“ f pw; χ, ηq “
logpw ´ xqdµpxq ´
logpw ´ xqdx,
nÑ8
χ`η´1
R
(1.3.14)
pnq
pnq
for all w P CzR. Note that since |βi ´ βi`1 | ě 1{n for all i “ 1, 2, , , n ´ 1, and
µn is a positive measure, the limit measure µ must satisfy 0 ď µ ď λ. Thus, in
particular µ is absolutely continuous with respect to the Lebesgue measure. We
will denote the class of all positive Borel measures on R with compact support such
that µ ď λ and }µ} “ 1 by Mλc,1 pRq.
If we formally replace the functions fn,1 and fn,2 in the integrand by the limit
function f we get the double contour integral
˛
˛
enpf pw;χ1 ,η1 q´f pz;χ2 ,η2 qq
1
pnq
dz
dw
LΓn γn ppχ1 , η1 q, pχ2 , η2 qq “
2
1
p2πiq n1 Γn
w´z
n γn
(1.3.15)
pnq
It is reasonable to assume that the asymptotics of JΓn γn is controlled by the asymppnq
totics of LΓn γn , i.e., by the limit function f . In which way this is true will depend
on where one is looking and have to be specified in each individual case.
For now we will consider f pw; χ; ηq. Let a “ min supppµq and b “ max supppµq.
For reasons that will become clear in a moment, we will only be interested in
deriving asymptotic scaling limits of (1.3.12) when pxi , yi q are in a neighborhood
of a point pχ, ηq P P, where P is the polygon
P “ tpχ, ηq P R2 : 0 ď η ď 1, χ ` η ´ 1 ě a, χ ď b, b ´ a ą 1u,
(1.3.16)
see figure 1.12.
pa, 1q
pb, 1q
P
pa ` 1, 0q
pb, 0q
Figure 1.12: The polygonal domain P of the parameter set.
For the existence of universal edge fluctuations of the frozen boundary, the
assumption that µn á µ as n Ñ 8 is not sufficient. Let
1
pnq
pnq
Pn “ tβ1 , β2 , ..., βnpnq u and Hn “ pZzPn q.
(1.3.17)
n
18
We will always assume the additional natural assumption that
lim dH pPn , supppµqq “ 0
and
nÑ8
lim dH pHn , supppλ ´ µqq “ 0,
nÑ8
(1.3.18)
where dH denotes the Hausdorff distance. With this additional assumption it also
follows that limnÑ8 min supppµn q “ limnÑ8 an “ a and limnÑ8 max supppµn q “
limnÑ8 bn “ b.
Note that due to the fact that f 1 pw; χ, ηq “ f 1 pw; χ, ηq, it is sufficient to only
consider roots in the upper-half plane H. Theorem 3.1 in [DM15a] imply that the
in fact, the function f pw; χ, ηq has at most a root in H whenever pχ, ηq P P.
We define the liquid region L to be the set of all pχ, ηq P P such that f 1 pw; χ, ηq
has one simple root in H. This condition with the reflection symmetry in the real
line gives the equations
" 1
f pw; χ, ηq “ 0
f 1 pw; χ, ηq “ 0.
Solving for χ and η as functions of w and w gives
$
’
pw ´ wqpeCpwq ´ 1q
’
& χ “ χL pwq “ w `
eCpwq ´ eCpwq
pw
´
wqpeCpwq ´ 1qpeCpwq ´ 1q
’
’
% η “ ηL pwq “ 1 `
,
eCpwq ´ eCpwq
where
ˆ
Cpwq “
R
dµpxq
w´x
(1.3.19)
(1.3.20)
is the Cauchy transform of µ. In [DM15a] we show that the map WL´1 : H Ñ L
defined by WL´1 pwq “ pχL pwq, ηL pwqq is in fact a homeomorphism.
Define the complex slope Ω “ Ωpχ, ηq P C by
Ωpχ, ηq “
WL pχ, ηq ´ χ
.
WL pχ, ηq ´ χ ´ η ` 1
(1.3.21)
The equations (1.3.19) imply that the complex slope satisfies the equation
˙´1
ˆ ˆ
1
p1 ´ ηqΩ
“ exp
χ`
´t
dµptq.
(1.3.22)
Ω
1´Ω
R
Note that by equations (1.3.21) and (1.3.22)
ˆ
dµptq
Ω “ exp
,
R t ´ WL pχ, ηq
(1.3.23)
and since WL pχ, ηq P H, it follows that ImrΩs ą 0 for all pχ, ηq P L. Moreover, by
differentiating (1.3.22) with respect to χ and η respectively, one see that Ω satisfies
the complex Burgers equation
Ω
BΩ
BΩ
“ ´p1 ´ Ωq .
Bχ
Bη
(1.3.24)
19
Using the complex slope Ω one define the Beta kernel BΩ : Z2 Ñ C, according
to:
1
BΩ pm, lq “
2πi
ˆ
Ω
p1 ´ zqm z ´l´1 dz,
(1.3.25)
Ω
where the integration contours are such that they cross p0, 1q Ă R when m ěˇ 0, and
p´8, 0q Ă R when m ă 0. It was shown in [Pet14], that if one lets µ “ λˇYm I ,
k“1 k
where Ik “ rak , bk s, and Ym
k“1 Ik is a disjoint union of intervals, then if one assumes
that
lim
nÑ8
1 pnq pnq
px , yi q “ pχ, ηq P L,
n i
for i “ 1, 2, .., r
and,
pnq
xi
pnq
´ xj
“ lij P Z
pnq
and yi
pnq
´ yj
“ mij P Z
are fixed whenever n is sufficiently large, then
pnq
pnq
pnq
pnq
lim ρr ppx1 , y1 q, px2 , y2 q, ..., pxrpnq , yrpnq q “ detrBΩ pmij , lij qsri,j“1
nÑ8
Though it is not done in this thesis, this result can be easily extended to the case
when µ P Mλc,1 pRq. In particular note that this implies that the macroscopic density
of particles are given by
ˆ Ω
dz
1
1
“ arg Ωpχ, ηq.
ρpχ, ηq “ BΩ p0, 0q “
2πi Ω z
π
Since the ultimate goal of this thesis is to characterize the possible universal
edge fluctuations at the boundary BL when such exist, an important subproblem,
is to characterize the geometry of the boundary BL of the liquid region for a given
µ P Mλc,1 pRq. Now how does one determine BL from the knowledge of WL´1 ? Since
R “ BH, it is natural to consider the image of sequences ω “ twn un Ă H, such that
limnÑ8 wn “ x for some x P R. More precisely, define
BLω pxq :“ tpχ1 , η 1 q P P : Dtwnk uk Ă twn un , lim WL´1 pwnk q “ pχ1 , η 1 qu.
kÑ8
1
We now observe that if ω “ twn un and ω 1 “ twm
um are two sequences in H such
1
that limnÑ8 wn “ limmÑ8 wm “ x and such that there exists integers N, M ą 0
1
such that wN `k “ wM
`k for all k ą 0, then BLω pxq “ BLω 1 pxq. Thus if we let the
conditions above define an equivalence condition among sequences, which we denote
by ω „ ω 1 , we see that the set BLω pxq only depends on ω through its equivalence
class rωs. If we let Sx denote the set of all equivalence classes of sequences ω
converging to x, we define
ď
BLpxq :“
BLrωs pxq.
ωPSx
20
Moreover, for every sequence twn un P H such that limnÑ8 |wn | “ `8,
"ˆ
˙*
ˆ
1
´1
lim WL pwn q “
`
xdµpxq, 0
:“ BLp8q :“ E8 .
nÑ8
2
R
(1.3.26)
It is not hard to see that the boundary BL decomposes according to
˙
ďˆ ď
BL “ BLp8q
BLpxq .
xPR
Therefore, the problem of determining BL reduces to the problem of determining
the sets BLpxq for every x P R.
Geometry of E
We will now consider the problem of determining BLpxq. In doing this it will be
natural to decompose BH “ R into the sets R “ R Y Snt pµq, where
R “ ppRzsupppµqq Y Rzsupppλ ´ µqq˝ ,
and Snt pµq :“ RzR,
and where Snt pµq is called the non-trivial support of µ. Note that R is an open set
and that Snt pµq Ă supppµq is a closed possibly empty set.
First consider the case when x P R. It will be convenient to decompose the set
R further. Let
R :“ Rµ Y Rλ´µ Y R0 Y R1 Y R2 ,
(1.3.27)
where
• Rµ :“ tx P Rzsupppµq : Cpxq ‰ 0u.
• Rλ´µ :“ Rzsupppλ ´ µq.
• R0 “ tx P Rzsupppµq : Cpxq “ 0u
• R1 is the set of all x P BpRzsupppµqq X BpRzsupppλ ´ µqq for which there exists
an interval, I :“ px2 , x1 q, with x P I, px, x1 q Ă Rzsupppµq and px2 , xq Ă
Rzsupppλ ´ µq.
• R2 is the set of all x P BpRzsupppµqq X BpRzsupppλ ´ µqq for which there
exists an interval, I :“ px2 , x1 q, with x P I, px, x1 q Ă Rzsupppλ ´ µq and
px2 , xq Ă Rzsupppµq.
Note that Rµ Y R0 and Rλ´µ are open sets and and R1 Y R2 “ BpRzsupppµqq X
BpRzsupppλ ´ µqq are discrete isolated sets. Furthermore since µ is absolutely
continuous with respect to the Lebesgue measure, µ has a density ϕ. Using this
density the sets R0 YRµ is the set of all x P R such that there exists a neighborhood
21
U of x such that ϕ “ 0 a.e. on U , and the set Rλ´µ is the set of all x P R such
that ϕ “ 1 a.e. on U . With this decomposition of R we show in [DM15a] that
BLpxq “ tpχE pxq, ηE pxqqu
(1.3.28)
where
pχE pxq, ηE pxqq :“
$ ˆ
˙
´Cpxq
eCpxq ` e´Cpxq ´ 2
’
’ x` 1´e
,
1
`
’
’
’
C 1 pxq
C 1 pxq
’ ˆ
’
˙
x´x1 ´CI pxq
CI pxq
´CI pxq
2
1
&
1 ´ p x´x2 qe
p x´x
` p x´x
´2
x´x1 qe
x´x2 qe
x` 1
1
1 ,1 `
1
1
’
CI pxq ` x´x
´ x´x
CI1 pxq ` x´x
´ x´x
’
2
1
2
1
’
’
C
pxq
I
’
px, 1 ´ e
px ´ x2 qq
’
’
%
px ´ e´CI pxq px ´ x1 q, 1 ` e´CI pxq px ´ x1 qq
if x P Rµ Y R0
if x P Rλ´µ
if x P R1
if x P R2
(1.3.29)
Above I :“ px2 , x1 q is any interval which satisfies x P I Ă Rzsupppλ ´ µq whenever
x P Rzsupppλ´µq, and the requirements of equation (1.3.27) whenever x P R1 YR2 .
´
Also, C is the Cauchy transform of equation (1.3.20), and CI pxq :“ RzI dµptq
x´t for all
x P I. It follows from above that WE´1 p¨q :“ pχE p¨q, ηE p¨qq : R Ñ BL is the unique
continuous extension, to R, of WL´1 p¨q “ pχL p¨q, ηL p¨qq : H Ñ L. In [DM15a] we
show that the extension is injective, and we define the edge, WE´1 pRq :“ E Ă BL,
as the image space of the extension. In fact, WE´1 P C ω pR, Eq is a real analytic
bijective parametrization of E.
The decomposition (1.3.27) induces a decomposition of E according to:
E “ Eµ Y Eλ´µ Y E0 Y E1 Y E2 ,
(1.3.30)
where Eµ “ WE´1 pRµ q, Eλ´µ “ WE´1 pRµ q, E0 “ WE´1 pR0 q, E1 “ WE´1 pR1 q, and
E2 “ WE´1 pR2 q. As we will see later, this decomposition is intimately connected
with the possible types of universal edge fluctuations of E. One can show that
for any sequence tpχn , ηn qun Ă L, such that limnÑ8 pχn , ηn q “ pχE , ηE q P E, the
boundary value of the complex slope Ω exists and equals
$ ´Cpxq
e
PR
if pχE , ηE q P Eµ
’
’
’
x´x2 ´CI pxq
’
P
R
if
pχE , ηE q P Eλ´µ
e
& x´x1
lim Ωpχn , ηn q “
(1.3.31)
1
if pχE , ηE q P E0
nÑ8
’
’
’
0
if pχE , ηE q P E1
’
%
8
if pχE , ηE q P E2
where x “ WE pχE , ηE q, and where limnÑ8 Ωpχn , ηn q “ 8 is viewed as a limit
on the Riemann sphere C Y t8u. Since limnÑ8 Ωpχn , ηn q “ limnÑ8 Ωpχn , ηn q,
we may view E as a shock of the complex Burgers equation (1.3.24). We will write
ΩE pχ, ηq for the boundary value of the complex slope when pχ, ηq P E. We note that
22
x´χ
ą 0 when pχ, ηq P Eµ and ΩE pχ, ηq ă 0 when pχ, ηq P Eλ´µ .
ΩE pχ, ηq “ x´χ´η`1
In particular, this implies that x P Rzpsupppµq Y rχ ` η ´ 1, χsq whenever x P Rµ
and x P pχ ` η ´ 1, χq whenever x P Rλ´µ . Using this observation together with
(1.3.29), it is easy to see that the function f 1 pw, χ, ηq, with pχ, ηq P E is complex
analytic in a neighborhood of x P R. To simplify notation we define
pkq
fEpxq pwq :“
dk
f pw; χE pxq, ηE pxqq.
dwk
(1.3.32)
It is in fact easy to see that the equations
p1q
fEpxq pxq “ 0
(1.3.33)
p2q
fEpxq pxq
(1.3.34)
“0
are equivalent to the parametrization for all x P RzpR0 Y R1 Y R2 q. Thus except
p1q
for the discrete set R0 Y R1 Y R2 the function fEpxq pwq has at least a double root
at x.
We will now turn to a more detailed description of the geometry of E. To
simplify notation we will write γE pxq :“ WE´1 pxq. A computation gives
γE1 pxq “ pχ1E pxq, ηE1 pxqq “ χ1E pxqp1, 1 ´ eCpxq q
ˆ Cpxq
˙
e
´1
p3q
“ ´ Cpxq 1 2 fEpxq pxqp1, 1 ´ eCpxq q.
e
C pxq
(1.3.35)
(1.3.36)
for all x P Rµ Y R0 , and
ˆ
γE1 pxq
“´
x´x1 CI pxq
´1
x´x2 e
x´x1 CI pxq
1
1
pCI pxq ` x´x
x´x2 e
1
˙
ˆ
˙
x ´ x1 CI pxq
p3q
f
pxq 1, 1 ´
e
.
1
x ´ x2
´ x´x
q2 Epxq
2
(1.3.37)
for all x P Rλ´µ . Similarly, one can show that
#
p2q
´2px ´ x2 qeCI pxq fEpxq pxqp0, 1q x P R1
1
γE pxq :“
p2q
2px ´ x1 qeCI pxq fEpxq pxqp1, ´1q x P R2
(1.3.38)
p3q
One easily sees that γE1 pxq “ 0 iff and only if fEpxq pxq “ 0 whenever x P RzpR1 Y
p2q
R2 q and γE1 pxq “ 0 iff and only if fEpxq pxq “ 0 whenever x P R1 Y R2 . Furthermore,
p4q
p3q
in [DM15a], we show that fEpxq pxq ‰ 0 for all x P RzpR1 YR2 q whenever fEpxq pxq “ 0
p3q
and fEpxq pxq ‰ 0 for all x P R1 Y R2 . Let
!
)ď!
)
p3q
p2q
Rreg :“ x P RzpR1 Y R2 q : fEpxq pxq ‰ 0
x P R1 Y R2 : fEpxq pxq ‰ 0
23
and
Rsing :“ RzRreg .
In particular, it then follows that the unit tangent vector
tE pxq “
γE1 pxq
}γE1 pxq}
is continuous for all x P Rreg and limx1 Ñx tE pxq “ ´ limx1 Ñx tE pxq whenever x P
x1 ąx
x1 ăx
Rsing . Since the parametrization WE´1 “ γE : R Ñ E is injective, all regular points
of E, i.e. those points for which the E has a tangent equals E reg :“ WE´1 pRreg q.
Furthermore, the set E sing :“ WE´1 pRsing q is the set of singular points of E. In fact
as we shall see, the set E sing are cusp points of E.
Define the unit normal vector nE pxq to be
nE pxq “
t1E pxq ´ xt1E pxq, tE pxqytE pxq
}t1E pxq ´ xt1E pxq, tE pxqytE pxq}
Recall the extrinsic curvature
κE pxq “
detpγE1 , γE2 q
}t1 pxq ´ xt1E pxq, tE pxqytE pxq}
detptE pxq, nE pxqq.
“ E
1
3
}γE }
}γE1 pxq}
In [DM15a], we show that κE pxq ă 0 for all x P Rztx P R : χ1E pxq “ 0u. Since the
sign of κE equals the sign of detptE pxq, nE pxqq, this shows that the unit normal vector always points into the liquid region L when we are at a regular point on E. This
places strong restrictions on the geometry of E. For example, if we Taylor expand
γE psq around a regular point γE px0 q of E, using an local arc length parametrization,
then
γE psq “ γE px0 q ` stE pxq `
s2
|κE pxq|nE pxq ` ops2 q,
2
showing that the edge E is locally a parabola at every regular point.
We now consider the singular points of E where γE1 pxc q “ 0 and tE pxc q is discontinuous. We will only consider the case when xc P Rsing X Rµ as the other cases are
completely analogous. Let t´
tE pxq and nE` pxq :“ limxÑx`
nE pxq.
E pxq :“ limxÑx´
c
c
´
Then tE pxq points into the liquid region. A Taylor expansion shows that there
exists constants apxc q ă 0 and bpxc q ą 0 such that
3 `
4
γE pxq “ γE pxc q ` apxc qpx ´ xc q2 t´
E pxq ` bpxc qpx ´ xc q nE pxq ` Op|x ´ xc | q.
With u “ apxc qpx ´ xc q2 and v “ bpxc qpx ´ xc q3 we see that u and v satisfy the
bpxc q2 3
algebraic equation v 2 “
u , which has a cusp at p0, 0q. This concludes the
apxc q3
discussion about the geometry of E.
24
Geometry of BLzE
We now turn to the problem of determining BLpxq when x P Snt pµq. It turns out
that this problem is very difficult in general, and in [DM15b] we obtain a partial
characterization of BLpxq for subsets of Snt pµq. To study the image WL´1 pwn q of
limits of sequences twn un P H such that limnÑ8 wn “ x, it is useful to write
w “ u ` iv and Cpwq “ πHv ϕpuq ´ iπPv ϕpuq, where
ˆ
pu ´ tqϕptqdt
1
π R pu ´ tq2 ` v 2
ˆ
1
vϕptqdt
Pv ϕpuq :“
,
π R pu ´ tq2 ` v 2
Hv ϕpuq :“
(1.3.39)
(1.3.40)
where we recall that Pv ϕpuq is the Poisson integral of ϕ. Using (1.3.39) and (1.3.40),
(1.3.19) becomes
WL´1 pu, vq “ pχL pu, vq, ηL pu, vqq
ˆ
˙
e´πHv f puq ´ cospπPv f puqq
eπHv f puq ` e´πHv f puq ´ 2 cospπPv f puqqq
“ u`v
,1 ´ v
.
sinpπPv f puqq
sinpπPv f puqqq
(1.3.41)
Using (1.3.39), (1.3.40) and (1.3.41), it is easy to show the following. If x P Snt pµq
and there exists an ε ą 0 for which one of the following cases is satisfied:
1. suptPpx´ε,x`εq ϕptq ă 1 and inf tPpx´ε,x`εq ϕptq ą 0.
2. suptPpx´ε,xq ϕptq ă 1, inf tPpx´ε,xq ϕptq ą 0 and ϕpxq “ 0 for all t P px, x ` εq,
3. suptPpx´ε,xq ϕptq ă 1, inf tPpx´ε,xq ϕptq ą 0 and ϕpxq “ 1 for all t P px, x ` εq,
4. suptPpx,x`εq ϕptq ă 1, inf tPpx,x`εq ϕptq ą 0 and ϕpxq “ 0 for all t P px ´ ε, xq,
5. suptPpx,x`εq ϕptq ă 1, inf tPpx,x`εq ϕptq ą 0 and ϕptq “ 1 for all t P px ´ ε, xq,
then BLpxq “ tpx, 1qu. If on the other hand none of assumptions 1-5 is satisfied,
then things are much more subtle.
We will call a point x P Snt pµq generic if BLpxq “ tpx, 1qu and denote the set
gen
of generic points by Snt
pµq. Otherwise x will be called singular, and the set of
sing
singular points will be denoted by Snt
pµq. Before proceeding, we will do a short
digression into harmonic analysis.
Let ϕ P L1loc pRq. The set of points such that
lim
hÑ0`
1
2h
ˆ
x`h
|ϕpxq ´ ϕptq|dt “ 0
x´h
25
is called the Lebesgue set of of ϕ, which we denote by Lϕ . It follows from a wellknown result in harmonic analysis that almost every point belongs to the Lϕ . Let
for all for k ą 0
Γk pxq :“ tpu, vq P H : v ą 0 and |u ´ x| ă kvu.
The cone Γk pxq is shown figure (1.13).
v
Γk pxq
|u ´ x| “ kv
u
x
Figure 1.13: Cone in H with tip at x P R
It is a well-known fact from harmonic analysis that if ϕ P Lp pRq for 1 ă p ď 8,
then for every k ą 0 and every sequence twn un Ă Γk pxq such that limnÑ8 wn “ x,
lim Pvn ϕpun q “ ϕpxq
nÑ8
(1.3.42)
for all x P Lϕ . Define the Hilbert transform of ϕ to be
ˆ
1
ϕptqdt
Hϕpxq :“ lim
,
`
εÑ0 π |x´t|ąε x ´ t
whenever the limit exits. One can show that in fact the limit exists for almost every
x and H : Lp pRq Ñ Lp pRq defines a bounded linear operator for every 1 ă p ă 8.
Furthermore, a theorem in [SW97] shows that
Pv pHϕqpuq “ Hv ϕpuq.
Thus, if ϕ P Lp pRq, 1 ă p ă 8, then if follows from (1.3.42) that for every k ą 0
and every sequence twn un Ă Γk pxq such that limnÑ8 wn “ x,
lim Hvn ϕpun q “ Hϕpxq
nÑ8
for all x P LHϕ .
Now, assume that ϕ is a density of a measure µ P Mλc,1 pRq, then since ϕ P
8
L pRq, and ϕ has compact support, it follows that ϕ P Lp pRq for all 1 ď p ď 8.
26
Thus, in view of the results on the convergence of the Poisson integral we will call
a point x P Snt pµq regular, if for every sequence twn un Ă H, such that there exists
a k ą 0 such that twn un Ă Γk pxq we have
lim pχL pwn q, ηL pwn qq “ px, 1q.
nÑ8
reg
pµq, and in [DM15b] we conjecture that
We denote the set of regular points by Snt
reg
gen
Snt pµq “ Snt pµq.
Thus far, we have not used the fact ϕ P L8 pRq implies additional consequences
for Hϕ besides belonging to Lp pRq. A function φ is said to be of bounded mean
oscillation if
ˆ x`h
1
}φ}BM O :“ sup sup
|φptq ´ M φpt, hq|dt ă 8,
(1.3.43)
xPR hą0 2h x´h
where
1
M φpx, hq :“
2h
ˆ
x`h
φptqdt.
(1.3.44)
x´h
The set of all such functions is denoted by BM OpRq, and one can prove that
BM OpRq is a Banach space. Moreover, a function φ P BM OpRq satisfy the JohnNirenberg inequality, which states that there exists constants c1 , c2 ą 0 such that
ˆ
˙
|tt P rx ´ h, x ` hs : |φptq ´ M φpt, hq| ą du|
d
ď c1 exp ´ c2
. (1.3.45)
2h
}φ}BM O
Finally, it follows from a well-known result that HpL8 pRqq Ă BM OpRq.
If one assumes Snt pµq˝ ‰ ∅ and that the density ϕ of µ satisfies the additional
regularity assumption that for all open intervals I Ă Snt pµq˝
0 ă λpI X tt P Snt pµq : 0 ă ϕptq ă 1uq ă 1,
(1.3.46)
then in [DM15b], we show using the additional control provided by the inequality
(1.3.45), that there exists a typical set G Ă Snt pµq˝ , such that for all open intervals
I Ă Snt pµq˝
0 ă λpI X Gq ă 1,
gen
and G Ă Snt
pµq. In particular, this shows that for all x P Snt pµq˝ , tpx, 1qu Ă
BLpxq. On the other hand in [DM15b], we identify four different classes of singular
points:
sing,I
• x P Snt
pµq if and only if there exists a δ ą 0 and a function ϕ : R Ñ R
for which x is in the Lebesgue set of ϕ, ϕpxq “ 0, |Hϕpxq| ă `8, and
f ptq “ χrx´δ,xs ptq ` ϕptq for almost all t.
27
sing,II
• x P Snt
pµq if and only if there exists a δ ą 0 and a function ϕ : R Ñ R
for which x is in the Lebesgue set of ϕ, ϕpxq “ 0, |Hϕpxq| ă `8, and
f ptq “ χrx,x`δs ptq ` ϕptq for almost all t.
´ f ptqdt
sing,III
• x P Snt
pµq if and only if R px´tq
2 ă `8 and Hf pxq ‰ 0.
sing,IV
• x P Snt
pµq if and only if
´
R
1´f ptqdt
px´tq2
ă `8 and Hp1 ´ f qpxq ‰ 0.
Moreover we show that every non-tangential limit exists for these classes of singular
points and is different from px, 1q. In particular this shows that singular points are
not regular points. If in addition we make some additional technical assumptions on
the density ϕ, then we show that BLpxq is a straight line joining the non-tangential
limit with the point px, 1q. Furthermore, we show that there exists a µ P Mλc,1 pRq
sing
such that λpSnt
pµq X Snt pµq˝ q ą 0. We also show that there exists a µ P Mλc,1 pRq
sing
such that Snt pµq is dense in a non-empty interval in Snt pµq˝ . In particular this
shows that the boundary BL can be very complicated and that there is probably
no complete characterization of BL for every µ P Mλc,1 pRq.
sing
We conjecture that the set Snt
pµq is a meagre set. In addition we leave as
an open problem whether our classes of singular points completely characterize of
sing
sing
sing,I
sing,II
sing,II
sing,IV
Snt
pµq, i.e., whetherSnt
pµq “ Snt
pµqYSnt
pµqYSnt
pµqYSnt
pµq.
We now consider the boundary behaviour of the complex slope Ω for sequences
tpχn , ηn qun Ă L such that limnÑ8 pχn , ηn q “ pχ, ηq P BLzE. First consider the case
gen
pµqu. One can show that almost all non-tangential
when pχ, ηq P tpx, 1q : x P Snt
limits exists and
lim Ωpχn , ηn q “ e´πHf pxq`iπf pxq P C.
nÑ8
Thus, such limit is thus not in general real, which should be contrasted to the case
sing
when pχ, ηq P E. On the other hand, if we assume that x P Snt
pµq, and that in
addition x is an isolated singular point, then for all sequences tpχn , ηn qun Ă L such
that limnÑ8 pχn , ηn q “ pχ, ηq P BLpxq we get
$ ´πHf pxq
e
PR
’
’
&
´eπHp1´f qpxq P R
lim Ωpχn , ηn q “
nÑ8
’
0
’
%
8
if
if
if
if
sing,III
x P Snt
pµq
sing,IV
x P Snt
pµq
sing,I
x P Snt
pµq
sing,II
x P Snt
pµq
(1.3.47)
This shows that at least a subset of BLsing are shocks of the complex Burgers
equation in the same way as E. For more on the complex Burgers equation and its
relation to dimer models see [KO07] and [KO05].
The fact that Ω can have complex valued boundary values for sequences tpχn , ηn qun Ă
L such that limnÑ8 pχn , ηn q “ px, 1q are natural from the point of view of tiling
models as described in section 2.2. In those models the measure µ is a solution to
a variational problem described in section 4. In particular for such a measures µ,
28
the support is a finite union of intervals and the density ϕ is Hölder continuous and
real analytic in the interior of the support. Furthermore, one can show that “genergen
ically”, for such a measure Snt pµq “ Snt
pµq. In particular, for every sequence
tpχn , ηn qun Ă L such that limnÑ8 pχn , ηn q “ pχ, ηq P BLzE and every x P Snt pµq
lim Ωpχn , ηn q “ e´πHf pxq`iπf pxq P C.
nÑ8
Furthermore, the boundary values Ωpx, 1q define a function Ω P C 1{2 pSnt pµq, Hq
and Ω P C ω pSnt pµq˝ , Hq. We may therefore analytically continue the complex slope
Ω into the liquid region of the other half of the polygon as depicted in figure 1.14.
Here note that we have made the inverse coordinate transformation of the domain
to that we made in section 2.1, see figure 1.8.
Complex boundary values of Ω at the dashed line
L
E
Figure 1.14: Analytic continuation of the complex slope Ω to the entire liquid region
L of the original cut-corner hexagon
Universal Edge Fluctuations and the Classification of the
Universal Boundary Fluctuations of E
As was mentioned in the introduction of section 3.2, the different scaling limits of the
kernel Kn depend on the root structure of the asymptotic functions f . Therefore,
a classification of the possible universal scaling limits at the edge E can be reduced
1
to a classification of the possible number of roots of fEpxq
pwq at w “ x. We will
therefore sketch an argument for how this is done in the section below before stating
the (conjectured) different possible universal edge fluctuations of E. In this thesis,
we only derive the Cusp-Airy case, however similar arguments can be applied to
the other cases as well.
29
Root Structure of the Function f pw; χ, ηq
Before discussing the possible types of universal edge fluctuations admitted by E,
we will discuss the root structure of f pw; χ, ηq for pχ, ηq P E. Define the signed
measure
ˇ
ˇ
ˇ
ν :“ µˇra,χ`η´1s ´ pλ ´ µqˇrχ`η´1,χs ` µˇrχ,bs ,
(1.3.48)
and let ν “ ν ` ´ ν ´ be the Hahn decomposition of ν. Since µ|Rλ´µ “ λ,
supppνq “ pra, χ ` η ´ 1s X supppµqq Y prχ ` η ´ 1, χszRλ´µ q Y prχ, bs X supppµqq.
(1.3.49)
Define
S1` “ ra, χ ` η ´ 1s X supppµq,
S ´ “ rχ ` η ´ 1, χszRλ´µ ,
and
S2` “ rχ, bs X supppµq.
(1.3.50)
Also denote
a1 “ mintS1` u, b1 “ maxtS1` u, a2 “ mintS ´ u, b2 “ maxtS ´ u,
a3 “ mintS2` u,
and b3 “ maxtS2` u.
Using the signed measure ν, we can write (1.3.14) as
ˆ
f pw; χ, ηq “
logpw ´ tqdνptq,
(1.3.51)
(1.3.52)
R
and we see that
ˆ
f 1 pw; χ, ηq “
R
dνptq
,
w´t
(1.3.53)
is holomorphic in Czsupppνq. We may similarly, for i “ 1, 2 define the signed
measures
ˇ
ˇ
ˇ
νn,1 :“ µn ˇra ,x {n`y {n´1q ´ pλn ´ µn qˇrx1 {n`y1 {n´1,x1 {ns ` µn ˇpx1 {n,bn s , (1.3.54)
1
ˇ n 1
ˇ
ˇ
νn,2 :“ µn ˇran ,x2 {n`y2 {n´1`1{nq ´ pλn ´ µn qˇrx2 {n`y2 {n´1`1{n,x2 {n´1{ns ` µn ˇpx2 {n´1{n,bn s ,
(1.3.55)
´
so that fn,i pw, xi {n, yi {nq “ R logpw ´ tqdνn,i ptq. It is an easy consequence of
Vitali’s theorem and the assumption (1.3.18) on the support of the empirical meapkq
sures that fn,i pw; xi {n, yi {nq for k ě 1 converges uniformly to f pkq pw; χ, ηq on
compact subsets of Czsupppνq whenever pxi {n, yi {nq Ñ pχ, ηq as n Ñ 8. Hence,
for every fixed compact set K Ă Czsupppνq, it follows from Rouché’s theorem that
1
fn,i
pw, xi {n, yi {nq and f 1 pw; χ, ηq has the same number of roots in K whenever
30
n is sufficiently large and f 1 pw; χ, ηq does not have a root wc P BK. Therefore,
when considering the possible number of roots of f 1 in Czsupppνq, we may consider
1
fn,1
pw, χ, ηq instead. Let
`
S1,n
“ ran , χ ` η ´ 1q X supppµn q,
Sn´ “ prχ ` η ´ 1, χq X
1
ZqzRλn ´µn
n
`
S2,n
“ rχ, bn s X supppµn q,
where Rλn ´µn :“
and,
(1.3.56)
1
n Zzsupppλn
´ µn q. Furthermore, let
`
`
`
`
S1,n
“ tp`
1,k : p1,k ă p1,k`1 , k P t1, 2, ..., |S1,n |uu
´
´
´
Sn´ “ tp´
k : pk ă pk`1 , k P t1, 2, ..., |Sn |uu
`
`
`
`
S2,n
“ tp`
2,k : p2,k ă p2,k`1 , k P t1, 2, ..., |S2,n |uu.
1
Then fn,1
is a rational approximation of f 1 . In particular,
ÿ
1
fn,1
pw; χ, ηq “
`
|u
kPt1,2,...,|S1,n
“
`
`
´
YSn
YS2,n
pPS1,n
:“
ÿ
´
kPt1,2,...,|Sn
|u
1
`
w ´ p´
k
ÿ
`
|u
kPt1,2,...,|S2,n
1
w ´ p`
2,k
ˆ
1
ź
1
´
w ´ p`
1,k
ÿ
pw ´ pq
p1pPS `
`
1,n YS2,n
ź
´ 1pPSn´ q
´
`
`
YSn
YS2,n
pPS1,n
˙
pw ´ p1 q
p1 ‰p
`
`
´
YSn
YS2,n
p1 PS1,n
Pn pw; χ, ηq
.
Qn pw; χ, ηq
1
Thus, tw P Czsupppνn,1 q : fn,1
pw; χ, ηq “ 0u “ tw P C : Pn pw; χ, ηq “ 0u, where we
`
`
note that degpPn pw; χ, ηqq ď |S1,n
| ` |Sn´ | ` |S2,n
| ´ 1. On the other hand, if x P
`
`
1
1
ppi,k , pi,k`1 q Ă R, then limxÑp` fn,1 px; χ, ηq “ ´8 and limxÑp` fn,1
px; χ, ηq “
i,k`1
i,k
`
`8. This implies that Pn pw; χ, ηq has at least one zero in the interval pp`
i,k , pi,k`1 q.
´ ´
Similarly, one sees that Pn pw; χ, ηq has at least one zero in the interval ppk , pk`1 q.
`
`
This implies that Pn pw; χ, ηq has at least |S1,n
|`|Sn´ |`|S2,n
|´3 real zeros. Together
with the bound on the degree on Pn this implies that for pχ, ηq P Pn :
`
`
1
• fn,1
pw; χ, ηq has at most |S1,n
| ` |Sn´ | ` |S2,n
| ´ 1 roots.
`
`
1
• fn,1
pw; χ, ηq has at least |S1,n
| ` |Sn´ | ` |S2,n
| ´ 3 real roots.
`
1
• fn,1
pw; χ, ηq has at most three real roots in the interval pp`
i,k , pi,k`1 q for k P
`
t1, 2, ..., |Si,n | ´ 1u.
´
1
• fn,1
pw; χ, ηq has at most three real roots in the interval pp´
k , pk`1 q for k P
´
t1, 2, ..., |Sn | ´ 1u.
1
• fn,1
pw; χ, ηq has at most two real roots in the interval pp|S ` | , p´
1 q.
1,n
31
1
• fn,1
pw; χ, ηq has at most two real roots in the interval pp|Sn´ | , p`
2,1 q.
1
• fn,1
pw; χ, ηq has at most two real roots in the interval p´8, p`
1,1 q.
1
• fn,1
pw; χ, ηq has at most two real roots in the interval pp|S ` | , `8q.
2,n
The rational approximation now shows that:
1. f 1 pw; χ, ηq has at most two roots in CzR occurring as conjugate pairs.
2. f 1 pw; χ, ηq has at most one triple or double root in Rµ Y Rλ´µ .
3. f 1 pw; χ, ηq has at most one double root in R1 Y R2 .
4. f 1 pw; χ, ηq has at most three real roots in any open interval I such that I Ă
rai , bi szsupppµq, i P t1, 2, 3u.
5. f 1 pw; χ, ηq has at most two real roots in the interval pb1 , a2 q.
6. f 1 pw; χ, ηq has at most two real roots in the interval pb2 , a3 q.
7. f 1 pw; χ, ηq has at most two real roots in the interval p´8, a1 q.
8. f 1 pw; χ, ηq has at most two real roots in the interval pb3 , `8q.
Incidentally, the above also proves the injectivity of the mapping WE´1 : R Ñ
E. Namely, assume that there exists two x, x1 P R such that pχE pxq, ηE pxqq “
pχE px1 q, ηE px1 qq. However, this would immediately contradict 2. or 3.
Classification of the Universal Boundary Fluctuations of E
Recall the edge E and its decomposition E “ Eµ Y Eλ´µ Y E0 Y E1 Y E2 from the
previous section. One only expects to find universal edge fluctuations at E, and
not in general for the whole of BL. Since this depends on the root structure of the
function f 1 pw; χE pxq, ηE pxqq, it is important to characterize E in terms of this. In
[DM15a] and the previous discussion we showed that f 1 pw; χE pxq, ηE pxqq is analytic
at x. In addition we show that one has 9 possible cases.
1. If pχE pxq, ηE pxqq P Eµ XE reg , then f 1 pw; χE pxq, ηE pxqq has a root of multiplicity
2 at w “ x. Moreover x P Rµ zrχE pxq ` ηE pxq ´ 1, χE pxqs.
2. If pχE pxq, ηE pxqq P Eλ´µ X E reg , then f 1 pw; χE pxq, ηE pxqq has a root of multiplicity 2 at w “ x. Moreover x P Rλ´µ X Rreg X pχE pxq ` ηE pxq ´ 1, χE pxqq.
3. If pχE pxq, ηE pxqq P Eµ X E sing , then f 1 pw; χE pxq, ηE pxqq has a root of multiplicity 3 at w “ x. Moreover x P pRµ X Rsing qzrχE pxq ` ηE pxq ´ 1, χE pxqs.
4. If pχE pxq, ηE pxqq P Eλ´µ X E sing , then f 1 pw; χE pxq, ηE pxqq has a root of multiplicity 3 at w “ x. Moreover x P Rλ´µ X Rsing X pχE pxq ` ηE pxq ´ 1, χE pxqq.
32
5. If pχE pxq, ηE pxqq P E0 YE8 , then f 1 pw; χE pxq, ηE pxqq “ Cpxq , and f 1 pw; χE pxq, ηE pxqq
has a root of multiplicity 1.
6. If pχE pxq, ηE pxqq P E1 XE reg , then f 1 pw; χE pxq, ηE pxqq has a root of multiplicity
1. Moreover, x P R1 X Rreg and χE pxq “ x.
7. If pχE pxq, ηE pxqq P E2 XE reg , then f 1 pw; χE pxq, ηE pxqq has a root of multiplicity
1. Moreover, x P R2 X Rreg and χE pxq ` ηE pxq ´ 1 “ x.
8. If pχE pxq, ηE pxqq P E1 X E sing , then f 1 pw; χE pxq, ηE pxqq has a root of multiplicity 2. Moreover, x P R1 X Rsing and χE pxq “ x.
9. If pχE pxq, ηE pxqq P E2 XE reg , then f 1 pw; χE pxq, ηE pxqq has a root of multiplicity
2. Moreover, x P R2 X Rsing and χE pxq ` ηE pxq ´ 1 “ x.
Of course one must also show that for each of the cases above there exists a
µ P Mλc,1 pRq such that it is realized. However, it is easy to see that for case 1-7
there are plentiful of measures that realize the different cases. Also case 1 always
holds for all µ P Mλc,1 pRq since we always have Eµ X E reg ‰ ∅. Finally, in [DJM15],
we show that there are measures µ P Mλc,1 pRq that realize case 8-9.
We now turn to how this characterization of the root structure of the function
f 1 pw; χ, ηq for pχ, ηq P E suggests a classification of all possible universal edge fluctuations of the edge E. In the integral representation of the kernel (1.3.11), we do
not have the function f pw; χ, ηq, but the functions fn,1 pw; χ, ηq and fn,2 pz; χ, ηq.
Therefore for the existence of universal scaling limits one will need some suitable
assumptions on the convergence of fn,i to f as n Ñ 8 for i “ 1, 2. However, in
the discussion below we will not discuss which precise assumptions are needed in
each particular case, but only give the correct universal scaling limits whenever
correct assumptions are made. One should also note that even if the assumptions
for the existence of universal scaling limits are not met, there may still exist a
scaling limits, though in general this limit is not a universal scaling limit. What
we mean by this is that this limit will depend on how the sequence of empirical
n
1 ÿ
δ pnq converges to the measure µ. In particular, if we consider two
measures
n i“1 βi
n
n
1 ÿ
1 ÿ
sequences of top-line measure, µn “
δβ pnq and µ̃n “
δ pnq say, such that
n i“1 i
n i“1 β̃i
both sequences converges weakly to the same measure µ, we could still get different
scaling limits.
Case 1: The Airy Process
Assume that case 1 holds. Then there exists sequences tan un ,tbn un , tcn un , tpξ1n , τ1n qun ,
tpξ2n , τ2n qun , tpχnE pxq, ηEn pxqqun , ttnE pxqun and tnEn pxqun , such that
lim pan , bn , cn q “ pa, b, cq,
nÑ8
lim pξ1n , τ1n q “ pξ1 , τ1 q P R2 ,
nÑ8
lim pξ2n , τ2n q “ pξ2 , τ2 q P R2
nÑ8
33
lim pχnE pxq, ηEn pxqq “ pχE pxq, ηE pxqq,
lim tnE pxq “ tE pxq,
nÑ8
nÑ8
lim nEn pxq “ nE pxq
nÑ8
for some constants a, b, c ‰ 0, such that when we rescale according to
pnq
pnq
pnq
pnq
px1 , y1 q “ npχnE pxq, ηEn pxqq ` an n2{3 ξ1n tnE pxq ` bn n1{3 τ1n nEn pxq,
px2 , y2 q “ npχnE pxq, ηEn pxqq ` an n2{3 ξ2n tnE pxq ` bn n1{3 τ2n nEn pxq,
then
pnq
lim
nÑ8
pnq
wn px1 , y1 q
c n
pnq pnq n
wn px2 , y2 q
1{3
pnq
pnq
pnq
pnq
Kn ppx1 , y1 q, x2 , y2 qq “ KAi ppξ1 , τ1 q, pξ2 , τ2 qq,
for some suitable conjugation factor wn px, yq, where
KAi ppξ1 , τ1 q, pξ2 , τ2 qq “
ˆ
˙
1ξ1 ăξ2
pτ1 ´ τ2 q2
1
1
3
´a
exp ´
´ pξ2 ´ ξ1 qpτ1 ` τ2 q ` pξ2 ´ ξ1 q
4pξ2 ´ ξ1 q 2
12
4πpξ2 ´ ξ1 q
ˆ ˆ
3
3
2
2
2
2
1
1
dwdz
`
e 3 pw ´z q´ξ1 w `ξ2 z ´pτ1 ´ξ1 qw`pτ2 ´ξ2 qz
p2πiq2 LL LR
w´z
is the extended Airy kernel and the integration contours are shown in figure 1.15.
For a reference on the Extended Airy Process see [BK08].
LL
π
3
LR
0
π
3
Figure 1.15: Integration contours for the extended Airy kernel
Case 2: Id-The Airy Process
Assume that case 2 holds. Then there exists sequences tan un ,tbn un , tcn un , tpξ1n , τ1n qun ,
tpξ2n , τ2n qun , tpχnE pxq, ηEn pxqqun , ttnE pxqun and tnEn pxqun , such that
34
lim pξ1n , τ1n q “ pξ1 , τ1 q P R2 ,
lim pan , bn , cn q “ pa, b, cq,
nÑ8
lim pξ2n , τ2n q “ pξ2 , τ2 q P R2
nÑ8
lim pχnE pxq, ηEn pxqq “ pχE pxq, ηE pxqq,
nÑ8
nÑ8
lim tnE pxq “ tE pxq,
lim nEn pxq “ nE pxq
nÑ8
nÑ8
for some constants a, b, c ‰ 0, such that when we rescale according to
pnq
pnq
pnq
pnq
px1 , y1 q “ npχnE pxq, ηEn pxqq ` an n2{3 ξ1n tnE pxq ` bn n1{3 τ1n nEn pxq,
px2 , y2 q “ npχnE pxq, ηEn pxqq ` an n2{3 ξ2n tnE pxq ` bn n1{3 τ2n nEn pxq,
then
pnq
pnq
wn px1 , y1 q
cn n
nÑ8 w pxpnq , y pnq q
n 2
2
lim
1{3
`
pnq pnq
pnq pnq ˘
Id ´ Kn ppx1 , y1 q, x2 , y2 q “ KAi ppξ1 , τ1 q, pξ2 , τ2 qq,
for some suitable conjugation factor wn px, yq.
Case 3: The Pearcey Process
Assume that case 3 holds. Then there exists sequences tan un ,tbn un , tcn un , tpξ1n , τ1n qun ,
tpξ2n , τ2n qun , tpχnE pxq, ηEn pxqqun , ttnE pxqun and tnEn pxqun , such that
lim pan , bn , cn q “ pa, b, cq,
nÑ8
lim pξ1n , τ1n q “ pξ1 , τ1 q P R2 ,
lim pξ2n , τ2n q “ pξ2 , τ2 q P R2
nÑ8
lim pχnE pxq, ηEn pxqq “ pχE pxq, ηE pxqq,
nÑ8
nÑ8
lim tnE pxq “ t`
E pxq,
nÑ8
lim nEn pxq “ nE` pxq
nÑ8
`
1
for some constants a, b, c ‰ 0, and where t`
E pxq :“ limx Ñx tE pxq and nE pxq :“
limx1 Ñx nE pxq, such that when we rescale according to
x1 ąx
x1 ąx
pnq
pnq
pnq
pnq
px1 , y1 q “ npχnE pxq, ηEn pxqq ` an n1{2 ξ1n tnE pxq ` bn n1{4 τ1n nEn pxq,
px2 , y2 q “ npχnE pxq, ηEn pxqq ` an n1{2 ξ2n tnE pxq ` bn n1{4 τ2n nEn pxq,
then
pnq
lim
nÑ8
pnq
wn px1 , y1 q
c n
pnq pnq n
wn px2 , y2 q
1{4
pnq
pnq
pnq
pnq
Kn ppx1 , y1 q, x2 , y2 qq “ KP e ppξ1 , τ1 q, pξ2 , τ2 qq,
for some suitable conjugation factor wn px, yq, where
1ξ ąξ
KP e ppξ1 , τ1 q, pξ2 , τ2 qq “ a 1 2
exp
2πpξ1 ´ ξ2 q
ˆ
´
pτ1 ´ τ2 q2
2pξ1 ´ ξ2 q
˙
35
`
1
2πi
ˆ
ˆ
1
e 4 pw
Γz
4
´z 4 q` 21 pξ1 w2 ´ξ2 z 2 q´τ1 w`τ2 z
γw
dwdz
,
w´z
and the integration contours are shown in figure 1.16. For a reference on the Pearcey
Process see [OR07].
Γz
γw
0
π
4
Figure 1.16: Integration contours for the Pearcey kernel
Case 4: Id-The Pearcey Process
Assume that case 4 holds. Then there exists sequences tan un ,tbn un , tcn un , tpξ1n , τ1n qun ,
tpξ2n , τ2n qun , tpχnE pxq, ηEn pxqqun , ttnE pxqun and tnEn pxqun , such that
lim pξ1n , τ1n q “ pξ1 , τ1 q P R2 ,
lim pan , bn , cn q “ pa, b, cq,
nÑ8
lim pξ2n , τ2n q “ pξ2 , τ2 q P R2
nÑ8
lim pχnE pxq, ηEn pxqq “ pχE pxq, ηE pxqq,
nÑ8
nÑ8
lim tnE pxq “ t`
E pxq,
nÑ8
lim nEn pxq “ nE` pxq
nÑ8
`
1
for some constants a, b, c ‰ 0, and where t`
E pxq :“ limx Ñx tE pxq and nE pxq :“
limx1 Ñx nE pxq, such that when we rescale according to
x1 ąx
x1 ąx
pnq
pnq
pnq
pnq
px1 , y1 q “ npχnE pxq, ηEn pxqq ` cn n1{2 ξ1n tnE pxq ` cn n1{4 τ1n nEn pxq,
px2 , y2 q “ npχnE pxq, ηEn pxqq ` cn n1{2 ξ2n tnE pxq ` cn n1{4 τ2n nEn pxq,
then
pnq
pnq
wn px1 , y1 q
cn n
nÑ8 w pxpnq , y pnq q
n 2
2
lim
1{4
`
pnq pnq ˘
pnq pnq
Id ´ Kn ppx1 , y1 q, x2 , y2 q “ KP e ppξ1 , τ1 q, pξ2 , τ2 qq,
for some suitable conjugation factor wn px, yq.
Case 5-7: The GUE Corner Process
Assume that case 5-7 holds. Then there exists sequences tcn un , tξ1n un and tξ2n un
such that
lim cn “ c,
nÑ8
lim ξ1n “ ξ1 P R,
nÑ8
lim ξ2n “ ξ2 P R
nÑ8
36
for some constant c ‰ 0, such that when we rescale according to
pnq
pnq
pnq
pnq
px1 , y1 q “ npχE pxq, ηE pxqq ` cn n1{2 ξ1n tE pxq ` rnE pxq,
px2 , y2 q “ npχE pxq, ηE pxqq ` cn n1{2 ξ2n tE pxq ` snE pxq,
for fixed r, s P Z` , tE pxq P R2 and nE pxq P R2 , where Z` “ tn P Z : n ě 0u, then if
pχ, ηq P E0 Y E8
pnq
lim
nÑ8
pnq
wnY px1 , y1 q
c n
pnq pnq n
wnY px2 , y2 q
1{2
pnq
pnq
pnq
pnq
KnY ppx1 , y1 q, x2 , y2 qq “ KGC ppξ1 , rq, pξ2 , sqq,
for some suitable conjugation factor wnY px, yq, if pχ, ηq P E1 then
pnq
lim
nÑ8
pnq
wnB px1 , y1 q
c n
pnq pnq n
wnB px2 , y2 q
1{2
pnq
pnq
pnq
pnq
KnB ppx1 , y1 q, x2 , y2 qq “ KGC ppξ1 , rq, pξ2 , sqq,
for some suitable conjugation factor wnB px, yq, and if pχ, ηq P E2 then
pnq
lim
nÑ8
pnq
wnR px1 , y1 q
c n
pnq pnq n
wnR px2 , y2 q
1{2
pnq
pnq
pnq
pnq
KnR ppx1 , y1 q, x2 , y2 qq “ KGC ppξ1 , rq, pξ2 , sqq.
for some suitable conjugation factor wnR px, yq. Here, we recall that the correlation
kernel of the yellow tiles KnY coincides with the correlation kernel of the interlaced
particle system Kn , whereas the correlation kernels KnR and KnB are those of the red
and blue tiles respectively. These are given by the correlation kernel Kn through the
particle transformations (1.3.8). In addition, if tpχ, ηqu P E8 , then tE pxq “ p1, 0q
and nE pxq “ p0, 1q, and if tpχ, ηqu P E0 , then tE pxq “ p´1, 0q and nE pxq “ p0, ´1q,
and if tpχ, ηqu P E1 , then tE pxq “ p0, 1q and nE pxq “ p´1, 0q if x “ max R1 and
tE pxq “ p0, ´1q and nE pxq “ p1, 0q otherwise, and if tpχ, ηqu P E2 , then tE pxq “
?1 p1, ´1q and nE pxq “ ?1 p1, 1q if x “ min R2 and tE pxq “ ?1 p´1, 1q and nE pxq “
2
2
2
?1 p´1, ´1q otherwise. The GUE corner kernel is given by
2
pξ2 ´ ξ1 qs´r´1
KGC ppξ1 , rq, pξ2 , sqq “ ´1sąr 1ξ2 ąξ1 2s´r
ps ´ r ´ 1q!
ˆ
ˆ
2
dw wr w2 ´z2 `2zξ2 ´2wξ1
`
dz
e
,
s
p2πiq2 Γ0
L w´z z
where ξ1 , ξ2 P R and r, s P Z` , and where the contours are shown in figure 1.17.
For a reference on the GUE corner process also known as the GUE minor process
see [OR06] and [JN06].
37
L
Γ0
Figure 1.17: Contours for the GUE-minor kernel.
Case 8-9: The Cusp-Airy Process
Assume that case 8 or 9 holds. If x P R2 , then for fixed r, s P Z, and sequences
pnq
tξn un and tτn un such that limnÑ8 pξn , τn q “ pξ, τ q P R2 , let xc “ ntnχE pxqu and
assume the rescaling
$
`
˘
pnq
1
1{3
’
’ x1 “ nxc ` 2` r ´ c0 n ξn˘
’
&
pnq
y1 “ nyc ` 12 r ` c0 n1{3 ξn
`
˘
(1.3.57)
1
1{3
’ x2 “ nxpnq
τn
c ` 2 r ´ c0 n
’
’
`
˘
%
pnq
y2 “ nyc ` 12 r ` c0 n1{3 τn .
(A similar rescaling exists when x P R1 ). Then if x P R2 ,
pnq
pnq
2
2
wnR px1 , y1 q cn 1{3 R pnq pnq pnq pnq
n Kn ppx1 , y1 q, x2 , y2 qq “ KCA ppξ, rq, pτ, sqq
nÑ8 w R pxpnq , y pnq q 2
lim
n
for some suitable conjugation factor wnR px, yq, and if if x P R1 , then
pnq
pnq
2
2
wnB px1 , y1 q cn 1{3 B pnq pnq pnq pnq
n Kn ppx1 , y1 q, x2 , y2 qq “ KCA ppξ, rq, pτ, sqq
nÑ8 w B pxpnq , y pnq q 2
lim
n
for some suitable conjugation factor wnB px, yq (and a different scaling). For r, s P Z
and ξ, τ P R the Cusp-Airy kernel is defined by
pτ ´ ξqs´r´1
KCA ppξ, rq, pτ, sqq “ ´1τ ěξ 1sąr
ps ´ r ´ 1q!
ˆ
ˆ
1
1 wr 1 w3 ´ 1 z3 ´ξw`τ z
3
`
dz
dw
e3
,
2
p2πiq LL `Cout
w ´ z zs
LR `Cin
(1.3.58)
and where the contours are defined in figure 1.18.
38
LL
π
3
LR
Cout
y
Cin
0
x
π
3
Figure 1.18: Integration contours for the Cusp-Airy kernel.
One should note that the Cusp-Airy process is a special type of cusp situation
with different asymptotics than in the Pearcey process case. In particular, when
r “ s “ 0, then the Cusp-Airy kernel reduces to the Airy kernel.
Non-Universal Edge Fluctuations
What happens when we consider edge fluctuations in the case when pχ, ηq P BLzE?.
If we consider the roots of the asymptotic function f 1 pw, χ, ηq, then taking a sequence tpχn , ηn qun Ă L such that limnÑ8 pχn , ηn q “ pχ1 , η 1 q P BLzE, the corresponding roots tpwpχn , ηn q, pwpχn , ηn qqun will converge to px, 0q, where x P Snt pµq.
In this case one can no longer perform a steepest descent analysis. Though this
has not been the study of this thesis, the author believes that in general no scaling
limit exists unless additional assumptions on the sequence of empirical measures
are assumed. When such limits exits they will no longer be universal. Compare
with results in [Joh08]
1.4
Discrete Orthogonal Polynomial Ensembles and
Equilibrium Measures
In this section we discuss the variational problem (1.2.4) from section 2.2. For more
details the reader is referred to the upcoming paper [Dus16] which is not part of
this thesis. I have chosen to include this section in order to give a complete and
coherent picture.
39
Variational Problem
Recall the induced probability distribution (1.2.2)
pnq
pnq
pn rpx1 , x2 , ..., xpnq
n qs “
1
Zn
ź
pnq
pxi
pnq
´ xj q2
n
ź
pnq
wn pxi q,
i“1
1ďiăjďn
of tiles stemming from the lozenge polygon models described in section 2.2. As
was described earlier, this can be viewed as putting this probability distribution
on the top line configuration β pN q P Zn of particles in discrete interlacing models,
pnq
pnq
pnq
where the x1 ă x2 ă ... ă xn denotes the positions of the free particles of
pnq
pN q
pN q
pN q
β1
ă β2
ă ...βN . From now
Ş on we will omit the superscript pnq in xi ,
and only write xi . Let Σn :“ Z{n Σ be the lattice generated by the intersection
of Z{n and Σ. We will always assume that xpnq P Σnn be the ordered n-tuple
xpnq “ px1 , ..., xn q, with x1 ă x2 ă ... ă xn . Furthermore, we will let Cn “ Cn pΣq
denote the configuration space of all ordered n-tuples xpnq P Σnn . Finally we will
assume that
n
Ñq
N
as N Ñ 8, for some 0 ă q ď 1. Define the discrete weighted logarithmic energy En
of a configuration x1 ă x2 ă ... ă xn according to
En px1 , x2 , ..., xn q : “
n
n
ÿ
1 ÿ
1
log |xi ´ xj |´1 `
Vn pxi q
npn ´ 1q i‰j
n ´ 1 i“1
(1.4.1)
where Vn pxq :“ ´ n1 log wn pxq. Using the logarithmic energy En we can write (1.2.2)
as
pn rpx1 , x2 , ..., xn qs “
1 ´npn´1qEn px1 ,x2 ,...,xn q
e
,
Zn
where
Zn “
ÿ
e´npn´1qEn px
pnq
q
xpnq PCn
is the partition function. The logarithmic potential of finite positive Borel measure
µ is defined according to:
ˆ
µ
U pzq :“ log |z ´ t|´1 dµptq, z P C.
(1.4.2)
With VR “ Ypk“1 Ikr , VL “ Yqk“1 Ikl and
ź
wn pxq “
|x ´ yjr |
yjr PYk
Ir
j“1 j
ź
yjl PYk
Il
j“1 j
|x ´ yjl |,
40
one gets
lim ´
nÑ8
1
log wn pxq “ U χI r pxq ` U χI l pxq,
n
where χI r pxq “ χpYpk“1 Ikr q pxq and χI l pxq “ χpYqm“1 Im
l q pxq denote the characteristic
p
q
r
l
functions of the intervals pYk“1 Ik q and pYm“1 Im q respectively. We therefore define
the external potential V pxq to be
V pxq “ U χI r pxq ` U χI l pxq.
(1.4.3)
Let Mλ1 pΣq denote the space of all positive Borel measures ν such that }ν} “ 1,
ν ď λ and supppνq Ă Σ. Define the weighted logarithmic energy of a measure
ν P Mλ1 pΣq to be:
ˆ
ˆ
´1
IV rνs “
log |t ´ s| dνptqdνpsq `
V ptqdνptq for ν P Mλ1 pΣq. (1.4.4)
ΣˆΣ
Σ
From Thm. 2.1 in [DS97] we know that there exists a finite constant EVλ , called the
minimum energy, and a unique extremal measure µ :“ µλV P Mλ1 pΣq such that :
IV rµλV s “
inf
νPMλ
1 pΣq
tIV rνsu “ EVλ .
Moreover, since V pxq is continuous there exists a constant FVλ such that the following variational inequalities hold:
λ
2U µV pxq ` V pxq ě FVσ
µ
2U pxq ` V pxq ď
FVλ
holds everywhere on Sλ´µλ ,
V
holds everywhere on Sµλ .
V
(1.4.5)
(1.4.6)
where Sν “ supppνq denotes the support of a measure ν. Consequently,
λ
2U µV pxq ` V pxq “ FVλ
for all x P Sλ´µλ X Sµλ .
V
V
(1.4.7)
If in addition Sλ´µλ X Sµλ ‰ H, then the Lagrange multiplier FVλ is unique and
V
V
equals
FVλ “ maxtl P R : 2U µ pxq ` V pxq ě l holds pλ ´ µq-a.e. u
(1.4.8)
We will assume that this is always the case. These results motivate the following
definitions: Let
IS :“ pSµ zpSσ´µ X Sµ qq˝
IB :“ Sσ´µ X Sµ
IV :“ ΣzSµ˝
(Saturated region, constraint is active),
(1.4.9)
(Band, constraint is not active),
(1.4.10)
(Void, outside the support of µ).
(1.4.11)
41
If we suppose that the unconstrained extremal measure µV for the potential V
has density ψV (this is true for example when V is continuous on Σ, see [ST97]),
i.e.,
IV rµV s “ inf tIV rνsu “ EV ,
νPM1
then from [DS97], we have the following characterization of IS :
IS “ tx P Σ : ψV pxq ě 1u.
Hence, we may determine IS by first solving the unconstrained equilibrium problem.
However, we will not pursue this method further, but instead make an ansatz for
the structure of IS , IB and IV , as will be explained in the next section. Under
the assumptions that V is real analytic in the interior of Σ, it follows that the
constrained extremal measure µ has density dµ
dx “ ψpxq, such that where ψ is
Hölder continuous, see [DS97]. Hence, from the variational inequalities we have
ˆ
2
log |x ´ y|´1 ψpyqdy “ ´V pxq ` FVσ for x P IB .
Σ
Following the derivation in [Dei00], we multiply both sides by some φ1 for φ P
Cc8 pIB q and integrate, which gives
ˆˆ
˙
ˆ
ˆ
1
´1
2
φ pxq
log |x ´ y| ψpyqdy dx “
p´V pxq ` FVσ qφ1 pxqdx for x P IB .
IB
IB
Σ
Let
ˆ
log |x ´ y|´1 ψpyqdy.
F pxq “ 2
Σ
Then the right-hand side is the weak derivative of F and we get
ˆˆ
˙
ˆ
1
´1
DF pxq “2
φ pxq
log |x ´ y| ψpyqdy dx
IB
Σ
ˆ
˙
ˆ
ˆ
1
2
2
“ lim ´
φ pxq
logp|x ´ y| ` qψpyqdy dx by dominated convergence
Ñ0
IB
Σ
˙
ˆˆ
ˆ
2px ´ yq
ψpyqdy
dx
“ lim ´
φpxq
2
2
Ñ0
IB
Σ |x ´ y| ` q
ˆ
˙
ˆ
ˆ
ψpyq
“´2
φpxq p.v.
dy dx.
IB
Σ x´y
Thus, for all φ P Cc8 pΣq
ˆ
ˆ
ˆ
φpxq p.v.
IB
1
Σ
˙
ˆ
ψpyq
1
dy dx “
φpxqV 1 pxqdx
x´y
2 IB
Since V P CpIB q and ψ is Holder continuous we get
ˆ
ψpyq
1
p.v.
dx “ V 1 pxq for x P IB .
x
´
y
2
Σ
42
Large Deviation Principle and Convergence of the Empirical
Measure
In [Dus16], we show that the DOPE defined by (1.2.2) satisfy a large deviation
estimate. More precisely, let
Aεn :“ tx P Cn pΣq : En pxq ´ EVλ ą εn u.
Then, we show that for all 0 ă α ă 1 there exists a constant C independent of α
such that when εn “ n´α we have the estimate
PDOP E,n rAn´α s ď e´n
2´α
`Cn log n
.
In particular, this estimate shows that the empirical measure
νn “
n
1 ÿ
δ
n i“1 xi {n
converges weakly in probability to the constrained equilibrium measure µλV . In
turn, this proves that the empirical measure µn of the top line distribution β pnq
converges weakly in probability to the measure µ, defined according to
ˇ
µpXq “ λˇI l pqXq ` µλV pqXq
for every Borel subset X of R (if we are interested in the left side of the decomposed
polygon).
Scalar Riemann-Hilbert Problems
We will now turn to the problem of determining the density ψ of µλV . Assume that
Ťk
Σ “ i“1 rAi , Bi s. Since
V 2 pxq “
d2 U χI r pxq d2 U χI l pxq
`
“
dx2
dx2
ˆ
Ir
dt
`
px ´ tq2
ˆ
Il
dt
ą 0,
px ´ tq2
for all x P Σ,
V is convex on Σ. It follows from Theorem 2.16 in [DS97] that supppψq X pAi , Bi q
is an (possibly empty) interval for all i “ 1, 2, ..., k. This implies that the structure
of the support in pAi , Bi q can only be one of a finite number of possibilities. More
precisely, if the support has the structure void-band, we write V B, and if the
support has the structure saturated-void we write SV . One can show that the only
possibilities are V, B, S, BV, VB, SB, BS, VBV, BSB, SBS, VBS, SBV, BSBV,
VBSB, VBSBV. Therefore, the total number of possible structures of the support
equals 15k .
We will now show how one determines if an ansatz is the correct solution of
the variational problem and in addition determine ψ. Assume that IB X rAi , Bi s “
43
pai , bi q ‰ ∅ for all i “ 1, 2, ..., k. From the variational inequalities (1.4.5)-(1.4.6)
we have
ψptq “ 1 t P IS ,
ˆ
p.v.
Σ
1
ψpyq
dx “ ´ V 1 pxq x P IB .
y´x
2
Let for w “ u ` iv
ˆ
ψptqdt
“ ´πHv ψpuq ` iπPv ψpuq.
t´w
F pwq “
R
Then, by the Hölder continuity of ψ, it follows that
F` pxq :“
F´ pxq :“
lim
F pwq “ ´πHψpxq ` iπψpxq,
lim
F pwq “ ´πHψpxq ´ iπψpxq.
w“u`ivÑx
vą0
w“u`ivÑx
vă0
Thus,
1
pF` pxq ´ F´ pxqq “ ψpxq “ 1, for x P IS
2πi
1
pF` pxq ` F´ pxqq “ ´Hψpxq for x P IB .
2π
(1.4.12)
(1.4.13)
We would like to convert (1.4.13) into a difference. In order to do that we define
the sectionally analytic function Rpzq through the equation
Rpzq2 “
k
ź
pz ´ ai qpz ´ bi q z P ĈzΓ
i“1
and
lim
zÑ8
Rpzq
“ 1.
zk
More precisely, we let
˙
ˆ ÿ
k
pz ´ bi q
1
log
Rpzq “
pz ´ ai q exp
2 i“1
pz ´ ai q
i“1
k
ź
where log denotes the principal branch of the logarithm. Then, for x P paj , bj q
R˘ pxq “
k
ź
i“1
px ´ ai q exp
ˆ ÿ
˙
k
1
|x ´ bi |
πi
log
˘
2 i“1
|x ´ ai |
2
44
k
ź
“
px ´ ai q
i“1
k
ź
d
i“1
|x ´ bi | ˘ πi
e 2
|x ´ ai |
b
źa
“ ˘p´1qk´j i pbj ´ xqpx ´ aj q
px ´ bi qpx ´ ai q
i‰j
In particular,
R´ pxq “ ´R` pxq.
Moreover,
k
ź
ˆ ÿ
˙
k
1
|x ´ bi |
Rpxq “
px ´ ai q exp
log
2 i“1
|x ´ ai |
i“1
k
ź
“
sgnpx ´ ai q
i“1
k a
ź
px ´ bi qpx ´ ai q for x P RzIB .
i“1
Using Rpzq, we define the sectionally analytic functions
Gpzq “
F pzq
Rpzq
with branch cuts IB Y IS . The jump of Gpzq over IS Y IB equals
$ 2πi
’
if x P IS
’
’
& Rpxq
G` pxq ´ G´ pxq “
’
1
’
’ ´V pxq if x P I .
%
B
R` pxq
(1.4.14)
Thus, we seek a function G, analytic in CzpIB Y IS q with the jump condition
G` pxq ´ G´ pxq over IB Y IS prescribed by (1.4.14). This is a so called scalar
Riemann-Hilbert problem. Such problems do not have a unique solution, since if
G is a solution, then G1 “ G ` E is also a solution for every entire function E. A
solution is given by
ˆ
1
pG` pxq ´ G´ pxqqdx
,
Gpzq “
2πi IB YIS
x´z
which gives
ˆ
F pzq “ RpzqGpzq “ Rpzq
IS
dx
Rpzq
´
Rpxqpx ´ zq
2πi
ˆ
IB
V 1 pxqdx
.
R` pxqpx ´ zq
However, a Laurent series expansion of F pzq gives
ˆ
ˆ
8
ÿ
ϕptqdt
1
}ϕ}1
F pzq “
“´
tn ϕptqdt “ ´
` Op|z|´2 q,
n`1
z
z
R t´z
R
n“0
(1.4.15)
(1.4.16)
45
which shows that limzÑ8 F pzq “ 0. Imposing this normalization condition gives
a unique solution of the Riemann-Hilbert problem. On the other hand, a Laurent
series expansion of Gpzq gives
ˆˆ
˙
ˆ
8
ÿ
1
xn dx
1
xn V 1 pxqdx
F pzq “ ´Rpzq
´
z n`1
2πi IB R` pxq
IS Rpxq
n“0
Since limzÑ8 Rpzq{z k “ 1, this implies the moment equations
$ ˆ
ˆ
xn dx
1
xn V 1 pxqdx
’
’
´
“ 0 if n “ 0, 1, .., k ´ 1
’
’
2πi IB R` pxq
& IS Rpxq
ˆ
’
’
’
’
%
IS
xn dx
1
´
Rpxq
2πi
ˆ
IB
(1.4.17)
xn V 1 pxqdx
“1
R` pxq
if n “ k.
This gives k ` 1 equations on the 2k unknown parameters a1 , b1 , a2 , b2 , ..., ak , bk in
λ
the ansatz. Since 2U µV pxq ` V pxq “ FVλ for all x P IB , the variational inequalities
imply that
λ
λ
2U µV pbi q ` V pbi q “ 2U µV pai`1 q ` V pai`1 q
(1.4.18)
for all i “ 1, ..., k ´ 1. This gives the remaining k ´ 1 equations on the parameters.
However, even if an ansatz to the variational equations satisfies the system of equations (1.4.23) and (1.4.18) it is not necessarily true that it satisfies the variational
inequalities (1.4.5) and (1.4.6). We must therefore have
λ
λ
λ
λ
2U µV pxq ` V pxq ď 2U µV pai`1 q ` V pai`1 q,
x P IV
(1.4.19)
x P IS .
(1.4.20)
and
2U µV pxq ` V pxq ě 2U µV pai`1 q ` V pai`1 q,
Now if the ansatz satisfy this system of equalities and inequalities, then by conλ
struction we automatically have 2U µV pxq ` V pxq “ FVλ . Therefore by Theorem
2.1 in [DS97], any ansatz that satisfy (1.4.19)-(1.4.20) is the unique solution of the
variational systems. Therefore if the ansatz on the structure of the support of the
variational measure is correct, then the variational inequalities reduces to the finite dimensional system of equations and inequalities in a1 , b1 , ..., ak , bk . Using the
representation formula (1.4.15) and taking limits we get
ˆ
ˆ
dt
R` pxq
V 1 ptqdt
R` pxq
`
.
(1.4.21)
ψpxq :“
πi
2π
IS Rptqpt ´ xq
IB R` ptqpt ´ xq
So far we have not used the special structure of V 1 pxq. Let
ˆ
˙ ÿ
ˆ
˙
ˆ
p
q
ÿ
z ´ cri
pχI r pxq ` χI l pxqqdx
z ´ cli
W pzq “
“
log
.
`
log
x´z
z ´ dri
z ´ dli
R
i“1
i“1
46
Then W is an analytic function in CzpI r YI l q, and lim W pzq “ V 1 pxq˘πirχI r pxq`
zÑx˘
zPCzR
χI l pxqs. for x P I r Y I l . Consider the function W pzq{Rpzq analytic in CzpI r Y I l Y
IB q. Then Cauchy’s integral formula implies that
˛
1
W pwqdw
W pzq
“
,
Rpzq
2πi C Rpwqpw ´ zq
where C “ CR ` CIB ` CI r YI l is a positively oriented contour, as shown in figure
(1.19).
CR
C IB
CI r YI l
Figure 1.19: Integration contours.
Deforming CIB ` CI r YI l to the real axis gives
˛
ˆ
ˆ
W pzq
1
1
1
W pwqdw
V 1 pxqdx
2πipχI r pxq ` χI l pxqqdx
“
`
`
.
Rpzq
2πi CR Rpwqpw ´ zq πi IB R` pxqpx ´ zq 2πi I r YI l
Rpxqpx ´ zq
Using that
ˆ
lim
RÑ8
we get
Rpzq
πi
ˆ
IB
CR
W pwqdw
“ 0,
Rpwqpw ´ zq
V 1 pxqdx
“ W pzq ´
R` pxqpx ´ zq
ˆ
I r YI l
pχI r pxq ` χI l pxqqdx
,
Rpxqpx ´ zq
47
and hence
1
F pzq “ ´ W pzq ` Rpzq
2
ˆ
IS
Rpzq
dx
`
Rpxqpx ´ zq
2
ˆ
I r YI l
pχI r pxq ` χI l pxqqdx
.
Rpxqpx ´ zq
A similar computation shows that the moment equations (1.4.23) reduces to
$ ˆ
ˆ
xn dx 1
xn rχI r pxq ` χI l pxqsdx
’
’
`
“ 0 if n “ 0, 1, .., k ´ 1,
’
’
2 I r YI l
Rpxq
& IS Rpxq
ˆ
’
’
’
’
%
IS
xn dx 1
`
Rpxq
2
ˆ
I r YI l
xn rχI r pxq ` χI l pxqsdx
“1
Rpxq
if n “ k.
(1.4.22)
´ n dx
´
dx
We note that the integrals I xRpxq
and I Rpxqpx´zq
for I “ IS or I “ I r Y I l
are elliptic integrals if k “ 2 and hyperelliptic if k ą 2.
Parametrization of the Edge in the Cut-Corner Half Hexagon
Model
Consider the cut-corner half hexagon model. Let m be the length of the cut as
shown in figure 1.2. Looking at the red tiles gives I r “ I l “ r0, As with m “ qA
and 2 “ qB according to section 2.2 with q “ 1{m. Also Σ “ rA, Bs. We now make
the ansatsz that IV “ rA, as, IB “ ra, bs and IS “ rb, Bs. In addition we let α “ qa
and β “ qb. Since we are in the one-cut case, i.e. k “ 1, the moment equations
1.4.23 completely determines IB . The moment equations become
$ ˆ A
ˆ B
’
dx
dx
’
’
a
a
`
“0
& ´
px ´ aqpx ´ bq
px ´ aqpx ´ bq
ˆ0 A
ˆb B
(1.4.23)
’
xdx
xdx
’
’
a
a
´
`
“
1.
%
px ´ aqpx ´ bq
px ´ aqpx ´ bq
0
b
We have for z P H and for any interval J “ pc, dq, and such that pc, dq X pa, bq “ ∅,
ˆ
d
dt
a
“
pt
´
zq
pt
´ aqpt ´ bq
c
a
„
„ a
d
p pz ´ aqpz ´ bq ´ pt ´ aqpt ´ bqq2 ´ pt ´ zq2
1
´a
log
.
pt ´ zq
pz ´ aqpz ´ bq
c
This gives the Cauchy transform of µλV as
ˆ
F pzq “
a
B
dµλV ptq
t´z
48
a
˙ „
„ a
A
p pz ´ aqpz ´ bq ´ pt ´ aqpt ´ bqq2 ´ pt ´ zq2
z
´ log
“ log
z´A
pt ´ zq
0
a
„
„ a
B
2
p pz ´ aqpz ´ bq ´ pt ´ aqpt ´ bqq ´ pt ´ zq2
´ log
.
pt ´ zq
b
ˆ
This finally gives the Cauchy transform of the top line measure according to
ˆ
˙
ˆ 2
dµptq
w
Cpwq “
“ log
´ F pqwq.
w´m
0 w´t
Hence,
exppCpwqq
a
“a
‰
´zpb ´ zq p pz ´ aqpz ´ bq ´ pB ´ aqpB ´ bqq2 ´ pB ´ zq2
?
“a
‰
“
pB ´ zqpA ´ zq p pz ´ aqpz ´ bq ´ abq2 ´ z 2
a
“a
‰
p pz ´ aqpz ´ bq ´ pA ´ aqpA ´ bqq2 ´ pA ´ zq2
“a
‰
ˆ
p pz ´ aqpz ´ bqq2 ´ pb ´ zq2
a
‰
“a
´wpβ ´ wq p pw ´ αqpw ´ βq ´ p2 ´ αqp2 ´ βqq2 ´ p2 ´ wq2
“a
‰
“
?
p2 ´ wqpm ´ wq p pw ´ αqpw ´ βq ´ αβq2 ´ w2
a
“a
‰
p pw ´ αqpa ´ βq ´ pm ´ αqpm ´ βqq2 ´ pm ´ wq2
“a
‰
ˆ
.
p pw ´ αqpw ´ βqq2 ´ pβ ´ wq2
a
2
Let
´ βq. Furthermore, let c1 “
a Υ “ pw ´ αqpw ´aβq, so that Υ “ pw ´ αqpw
?
p2 ´ αqp2 ´ βq, c2 “ pm ´ αqpm ´ βq and c3 “ αβ. Now recall the equation
(1.3.22)
Ω´1 “ exppCpWL´1 pχ, ηqqq,
where WL´1 pχ, ηq “ χ ` p1´ηqΩ
. Recall that the edge E is a subset of tpχ, ηq P P :
Ω
ImrΩs “ 0u “ tpχ, ηq P P : ImrWL´1 s “ 0u. To simplify notation, we will write w
in place of WL´1 pχ, ηq. The equation for the complex slope Ω, or alternatively w,
becomes
“
‰“
‰
´wpw ´ βq pΥ ´ c1 q2 ´ p2 ´ wq2 pΥ ´ c2 q2 ´ pm ´ wq2
w´χ
“
‰“
‰
“
.
w´χ´η`1
p2 ´ wqpm ´ wq pΥ ´ c3 q2 ´ w2 Υ2 ´ pβ ´ wq2
(1.4.24)
Define the polynomials
“
‰“
‰
P1 pΥ, w, χ, ηq :“ pw ´ χqp2 ´ wqpm ´ wq pΥ ´ c3 q2 ´ w2 Υ2 ´ pβ ´ wq2
“
‰“
‰
` wpw ´ βqpw ´ χ ´ η ` 1q pΥ ´ c1 q2 ´ p2 ´ wq2 pΥ ´ c2 q2 ´ pm ´ wq2 .
49
and
P2 pΥ, wq :“ Υ2 ´ pw ´ αqpw ´ βq.
Then the equation for the complex slope becomes equivalent to the system of polynomial equations
"
P1 pΥ, w, χ, ηq “ 0
(1.4.25)
P2 pΥ, wq “ 0.
We can eliminate the auxiliary variable Υ by taking a resultant. Therefore, the
equation system (1.4.25) is reduces to
Qpw, χ, ηq :“ ResrP1 , P2 , Υs “ 0.
(1.4.26)
This shows that w and therefore also Ω satisfies an algebraic equation. The edge E
is a subset of tpχ, ηq P R2 : ∆pQ, wq “ 0u, where ∆ denotes the discriminant of Q.
This gives an implicit equation for E in χ and η as
ResrQ, Bw Q, ws “ 0.
1.5
(1.4.27)
Relation To Continuous Interlacing Model
Asymptotic Limit Shape
In this section we will compare the discrete interlacing model with the continuous interlacing model. We first introduce the continuous interlacing model. Let n ą 0 be
prq
prq
an integer and let for each r P t1, 2, ..., nu Cr be the set of all y prq “ py1 , ..., yr q P
prq
prq
prq
Rr such that y1 ą y2 ą ... ą yr . We will say that two configurations y prq P Cr
pr`1q
and y
P Cr`1 interlace if
pr`1q
y1
prq
pr`1q
ě y1 ě y2
pr`1q
ě ... ě yrprq ě yr`1 ,
and write y prq ď y pr`1q . Assume that the top line configuration β pnq “ y pnq is fixed
and consider the uniform probability distribution on C1 ˆ ... ˆ Cn´1 given by
dνry p1q , ...y pn´1q s “
1
δ pnq py pnq qdy pnq dy pn´1q ...dy p1q ,
Zn β
(1.5.1)
where Zn “ Zn pxpnq q “ VolpC1 ˆ ... ˆ Cn´1 q. For a reference on continuous interlacing systems, see [Met13] and [Def10]. In Proposition 2.4 in [Met13], Metcalfe derived the following contour intergral representation of the correlation kernel
pnq
pnq
KnR : prβn , β1 sˆt1, 2, ..., nuq2 Ă pRˆZq2 Ñ C of the determinantal point process
defined by (1.5.1):
KnR ppx1 , y1 q, px2 , y2 qq “
50
˛
˛
n ˆ
pnq ˙
pz ´ x2 qn´y1 ´1 1 ź w ´ βi
pw ´ x1 qn´y1 `1 w ´ z i“1 z ´ β pnq
Γn
γn
i
˛
˛
n ˆ
pnq ˙
n´y2 ´1
ź
1
1x1 ěx2 pn ´ y2 q!
w ´ βi
pz ´ x2 q
´
dz
dw
,
p2πiq2 pn ´ y1 ´ 1q! Γ1n
pw ´ x1 qn´y1 `1 w ´ z i“1 z ´ β pnq
1
γn
i
(1.5.2)
1x1 ăx2 pn ´ y2 q!
p2πiq2 pn ´ y1 ´ 1q!
dz
dw
where Γn is a positively oriented contour that contains the set tβ1n , ..., βnn uXrx2 , `8q
and nothing else of the set tβ1n , ..., βnn u, and passes through x2 , Γ1n is a positively
oriented contour that contains the set tβ1n , ..., βnn u X p´8, x2 q and nothing else of
the set tβ1n , ..., βnn u, and passes through x2 and γn contains x1 and the contours Γn
and Γ1n . The integration contours are shown in figure 1.20.
γn
Γ1n
Γn
an
x2
bn
pnq
Figure 1.20: Integration contours. Here, an “ βn
pnq
and β1
“ bn .
We note that the structural similarities between KnR and KnZ . In fact, we see
that we get KnR from KnZ by replacing the factor
śx2 ´1
pz ´ kq
pz ´ x2 qn´y2 ´1
2 `y2 ´n`1
śk“x
Ñ
.
x1
pw ´ x1 qn´y1 `1
k“x1 `y1 ´n pw ´ kq
Let
˛
˛
˙
n ˆ
pnq
pz ´ x2 qn´y2 ´1 1 ź w ´ βi {n
dz
dw
1
1
pw ´ x1 qn´y1 `1 w ´ z i“1 z ´ β pnq {n
n Γn
n γn
i
(1.5.3)
for all n P N. The integrand can be written as
pnq
JΓn γn ppx1 , y1 q, px2 , y2 qq
1
“
p2πiq2
exppnfn,1 pw; x1 , y1 q ´ nfn,2 pz; x2 , y2 qq
,
w´z
(1.5.4)
for all w, z P CzR, where
ˆ
pnq
pnq
fn,1 pw; χ, ηq “
logpw ´ tqdµn ptq ´ p1 ´ y1 {n ` 1{nq logpw ´ x1 {nq,
R
51
ˆ
pnq
fn,2 pw; χ, ηq “
pnq
logpw ´ tqdµn ptq ´ p1 ´ y2 {n ´ 1{nq logpw ´ x2 {nq.
R
If we now again assume that the empirical measure µn converges weakly to a limit
measure µ P Mc pRq, where Mc,1 pRq is the set of all positive Borel measures of
mass 1 and compact support, we define the limit function f “ f pw; χ, ηq as
ˆ
f pw; χ, ηq “
logpw ´ tqdµptq ´ p1 ´ ηq logpw ´ χq
(1.5.5)
R
ˆ
ˆ
“
logpw ´ tqdµptq ´
logpw ´ tqp1 ´ ηqδχ ptqdt
(1.5.6)
R
R
Assume that min supppµq “ a and max suppµ “ b and let
B “ ra, bs ˆ r0, 1s.
(1.5.7)
Define the liquid region L to be the set of all pχ, ηq P B such that f 1 pw; χ, ηq has a
unique root wc in H. A modification of the proof of Theorem 2.1 in [DM15a] shows
that this defines a homeomorphisms WL : L Ñ H, such that the inverse map is
given by
$
wCpwq ´ wCpwq
’
& χL pw, wq “
Cpwq ´ Cpwq
´1
(1.5.8)
WL pwq “
w´w
’
% ηL pw, wq “ 1 `
CpwqCpwq,
Cpwq ´ Cpwq
With w “ u ` iv one can also write (1.5.5) as
ˆ
˙
vHv µpuq
pHv µpuq2 ` Pv µpuq2 q
pχL pu, vq, ηL pu, vqq “ u ´
,1 ´ v
,
Pv µpuq
Pv µpuq
where
(1.5.9)
ˆ
1
vdµptq
,
π R pu ´ tq2 ` v 2
ˆ
pu ´ tqdµptq
1
.
Hv µpuq :“
π R pu ´ tq2 ` v 2
Pv µpuq :“
The case which most resembles the asymptotic limits of discrete interlacing
systems is when µ P Ma.c.
1,c pRq, i.e., when the limit measure is absolutely continuous
with respect to the Lebesgue measure. In that case, µ has a density ϕ P L1 pra, bsq.
Since ra, bs is a compact set, it follows that ϕ P Lp pra, bsq for all 1 ď p ă 8. Let
x P supppµq˝ . It is easy to see that if both mϕ pxq ă `8 and mHϕ pxq ă `8 and
ˆ
ϕptqdt
“ `8, then for all non-tangentially convergent sequences tpun , vn qun Ă
2
R px ´ tq
H to x we have
lim pχL pun , vn q, ηL pun , vn qq “ px, 1q.
(1.5.10)
nÑ8
52
Here,
1
hą0 2h
ˆ
x`h
mϕ pxq :“ sup
(1.5.11)
|ϕptq|dt
x´h
ˆ
is the Hardy-Littlewood maximal function. On the other hand if
R
then
lim pχL pun , vn q, ηL pun , vn qq ‰ px, 1q.
nÑ8
ϕptqdt
ă `8,
px ´ tq2
(1.5.12)
A more detailed study of BL for continuous interlacing system will be postponed
for future work.
2
2.1
Summary of Results
Paper A
In paper A we derive the correlation kernel (1.3.7). We define the liquid region
L and show that it is homeomorphic to the upper-half plane H. We also define
the edge E and describe its geometry, a concise version of which is given in section
3.2.1 in the introduction. Finally we prove the characterization of the edge given
in section 3.3.1 and 3.3.2 in the introduction.
2.2
Paper B
In paper B we give a partial characterization of the geometry of BLzE. A very
condensed summary of this is given in section 3.2.2 in the introduction. The main
results in paper B is a proof that a typical set G in the interior of the non-trivial
support is generic and a geometric characterization of certain classes of singular
parts of BL.
2.3
Paper C
In paper C we prove the existence of a universal scaling limit, the Cusp-Airy process,
at a cusp point pχ, ηq P pE1 Y E2 q Y E sing for uniformly random discrete interlacing
systems. This implies that the Cusp-Airy process exists as a universal scaling limit
of certain tiling models of Petrov type. We also show that up to certain natural
assumptions, the universal scaling limit at the cusp of the edge in the cut-corner
hexagon model is the Cusp-Airy process. See figure 1 in the introduction.
53
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