Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
The Cusp-Airy Process
Erik Duse. Joint with Kurt Johansson and Anthony Metcalfe
Workshop on Large Random Structures in Two Dimensions Paris,
January 16th´20th, 2017
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Our Model: Uniformly Random
Discrete Gelfand-Tsetlin
Patterns
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Discrete Gelfand-Tsetlin Patterns
p4q
p4q
x3
p4q
p3q
y2
y1
ď
ă
ă
p2q
y1
ă
p3q
y3
p4q
y2
ă
ă
p3q
“
p4q
y3
ď
x1
“
“
“
p4q
y4
p4q
x2
ď
p4q
x4
ď
§
pnq
pnq
pnq
For each n ě 1, fix x pnq P Zn with x1 ą x2 ą ¨ ¨ ¨ ą xn .
Consider all Gelfand-Tsetlin patterns, with particle positions in Z,
which satisfy an asymmetric interlacing constraint, and with the
particles on the top row in the x pnq positions:
ď
§
p2q
y2
y1
ď
ă
p1q
y1
§
Impose uniform distribution. Call this the uniformly random
discrete interlacing process.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Asymptotic assumptions
§
Assume that
n
1ÿ
δ pnq Ñ µ weakly as n Ñ 8,
n i“1 xi {n
where Supppµq Ă ra, bs compact.
§
Note, µ has density less than or equal to 1.
§
Additionally assume that µ is not Lebesgue measure on an interval
of length 1, a degenerate case.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Origin of the discrete process: Tilings of a ‘half-hexagon’
m`n
n
n
m
§
A ‘half-hexagon’ with sides of length n ě 1 and m ě 1. The dotted
line representing the upper boundary is considered to be ‘open’.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Three different types of lozenges
§
All sides are of lenght 1.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
An example tiling
8
8
9
§
§
Fix n vertical tiles on the upper boundary.
Impose the uniform probability measure on the set of all possible
tilings with the tiles on the upper boundary in these deterministic
positions.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Equivalent interlaced particle system.
8
8
9
§
Place particles in the center of each vertical tile. The particles on
the top row are deterministic, and the others are random. The set of
all particle positions completely determine the tiling.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Regular hexagon example.
8
8
4
§
In this case µ “ 21 λr0, 21 s ` 21 λr1, 23 s , where λ is Lebesgue measure.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Asymptotic behaviour of the discrete process
§
§
We showed that the discrete process is determinantal with a
correlation kernel Kn : pR ˆ t1, 2, . . . , nuq2 Ñ C and derived an
integral representation for it. This was also done independently by L.
Petrov in [9].
Rescaling vertical and horizontal by n1 , interlacing implies that the
bulk of the rescaled particles asymptotically lie in
tpχ, ηq P ra, bs ˆ r0, 1s : b ě χ ě χ ` η ´ 1 ě au:
pa, 1q
pb, 1q
pa ` 1, 0q
The Cusp-Airy Process
pb, 0q
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Local asymptotic behaviour
§
§
Fix pχ, ηq in this shape.
We examined the local asymptotic behaviour as n Ñ 8, in a
pnq
pnq
pnq
pnq
neighbourhood of pχ, ηq by considering Kn ppx1 , y1 q, px2 , y2 qq
pnq
pnq
pnq
pnq
where n1 px1 , y1 q Ñ pχ, ηq and n1 px2 , y2 q Ñ pχ, ηq.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Correlation kernel
§
Kn ppx1 , y1 q, px2 , y2 qq “ ny2 ´y1
ż
dw
cn
śx1 ´1
ż
dz
Cn
` n´y2 ˘
n
1
p2πiq2 Jn
j
j“x `y ´n`1 pz ´ n q
śy2 1 1
j
j“x2 `y2 ´n pw ´ n q
` Rn , where Jn equals
n
ź
1
w ´ z i“1
˜
and
– cn contains n1 tx2 ` y2 ´ n, x2 ` y2 ´ n ` 1, . . . , x2 u.
pnq
pnq
– Cn contains all of t n1 xj : xj ě x1 u, but none of
pnq
pnq
t n1 xj : xj ď x1 ` y1 ´ nu.
– cn and Cn do not-intersect, and cn contains Cn .
The Cusp-Airy Process
pnq
w ´ xi {n
pnq
z ´ xi {n
¸
,
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Integrand
§
Integrand equals
1
w ´z
exppnfn pw q ´ nf˜n pzqq, where
¸
˜
ˆ
˙
vn
pnq
n
ÿ
xi
1ÿ
1
j
fn pw q :“
´
log w ´
,
log w ´
n i“1
n
n j“v `s ´n
n
n
n
˜
¸
ˆ
˙
uÿ
pnq
n
n ´1
ÿ
x
1
1
j
log z ´ i
´
log z ´
.
f˜n pzq :“
n i“1
n
n j“u `r ´n`1
n
n
§
n
1
To first order, this approximates to w ´z
exppnfpχ,ηq pw q ´ nfpχ,ηq pzqq,
where
żb
żχ
fpχ,ηq pw q :“
logpw ´ xqµrdxs ´
logpw ´ xqdx.
a
χ`η´1
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Steepest descent analysis
§
We used steepest descent analysis to examine the asymptotic
behaviour: It should depend only on the roots of,
żb
1
pw q “
fpχ,ηq
a
żb
“
χ
µrdxs
´
w ´x
µrdxs
´
w ´x
żχ
χ`η´1
żχ
χ`η´1
dx
w ´x
pλ ´ µqrdxs
`
w ´x
ż χ`η´1
a
µrdxs
.
w ´x
§
λ is Lebesgue measure.
§
The cancellation between λ and µ on the interval rχ ` η ` 1, χs will
be important!
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Liquid region
§
Definition: The liquid region, L, is the set of all pχ, ηq for which
1
has a unique root in H :“ tw P C : Impw q ą 0u.
fpχ,ηq
§
Theorem: The map from L to the unique root in H is a
homeomorphism. (In [4])
§
We found the inverse homeomorphism, and used this to get a
complete description of BL for a broad class of µ.
§
Too many cases to list, so we go to examples.
§
Let ϕ : ra, bs Ñ r0, 1s represent the density of µ.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Example 1: ϕpxq “
1
2
for all x P r´1, 1s.
(−1, 1)
(1, 1)
L
E2
(0, 0)
E1
(1, 0)
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
The edge in example 1
§
1
has a unique root of
E1 is the set of all pχ, ηq for which fpχ,ηq
3
2
1
‰ 0).
multiplicity 2 in p1, `8q (fpχ,ηq “ fpχ,ηq “ 0 and fpχ,ηq
§
Theorem: The map from E1 to the unique repeated root is bijective,
and is an extension of the bulk homeomorphism.
§
The inverse is defined by t ÞÑ pχE ptq, ηE ptqq for all t P p1, `8q,
where
χE ptq :“ t `
e C ptq ´ 1
e C ptq C 1 ptq
and
ηE ptq :“ 1 `
pe C ptq ´ 1q2
,
e C ptq C 1 ptq
where C : CzSupppµq Ñ C is the Cauchy transform of µ.
§
Similarly for E2 and p´8, ´1q.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Example 2: ϕpxq “
(0, 1)
1
2
for all x P r0, 1s Y r2, 3s.
(1, 1)
(2, 1)
(3, 1)
L
(1, 0)
(3, 0)
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Example 3: ϕpxq “ 1 for all x P r0, 21 s Y r1, 32 s.
(0, 1)
( 12 , 1)
( 12 , 12 )
(1, 1)
L
(0, 0)
The Cusp-Airy Process
( 32 , 1)
( 32 , 12 )
( 32 , 0)
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Example 4: ϕpxq :“ ?
1 for all x P r0, 13 s Y r1, 43 s Y rc, c ` 31 s,
1
where c :“ 12
p23 ` 217q.
(0, 1) ( 13 , 1)
(1, 1) ( 43 , 1)
(c, 1) (c + 13 , 1)
( 31 , 23 )
L
(c + 13 , 0)
(1, 0)
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Example 4: Jump points of ϕpxq
§
At x “ 1{3 and x “ 4{3 the density ϕ jumps from 1 to 0 and at
x “ 1 and x “ c the density jumps from 0 to 1.
§
More generally, if t P Supppµq and there exists
an interval
ˇ
rx ´ δ, x ` δs, for some δ ą 0, such that ϕˇrx´δ,x`δs “ χrx´δ,xs ptq,
weˇ say that x P R1 . Similarly, if x is such that
ϕˇrx´δ,x`δs “ χrx,x`δs ptq, we say that x P R2 .
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Example 5: ϕpxq “
15
16 px
´ 1q2 px ` 1q2 for all x P r´1, 1s.
(−1, 1)
(1, 1)
(0, 0)
The Cusp-Airy Process
(1, 0)
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Example 5: Singular Parts of The Boundary
In Example 5 we note that the parameterization given by
t Ñ pχE ptq, ηE ptqq from r´8, ´1q Y p1, 8s Ñ BL does not parametrize
the entire part of
č
tpχ, ηq P pa, bq ˆ p0, 1q : a ă χ ` η ´ 1 ď χ ă bu BL
with a “ ´1 and b “ 1 in this example. The remaining part
BLsing :“ ptpχ, ηq P pa, bq ˆ p0, 1q : a ă χ ` η ´ 1 ď χ ă bu
we call the singular part of the boundary of the liquid region.
The Cusp-Airy Process
č
BLqzE,
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Example 5: Singular Parts of The Boundary
In [5], we show that the geometry of BLsing can be very complicated. In
fact we show that in general BL is not homeomorphic to S 1 , and we may
have H1 pBLsing q “ `8, where H1 denotes the one-dimensional
Hausdorff measure.
Moreover, we conjecture that one does not have any universal edge
behaviour on BLsing .
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Example: A simulation of a random lozenge tiling of the
cut-corner hexagon model. (Picture from Kenyon)
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Definition of the Cusp-Airy Kernel
For r , s P Z and ξ, τ P R we define the Cusp-Airy kernel by
pτ ´ ξqs´r ´1
KCA ppξ, r q, pτ, sqq :“ ´1τ ěξ 1sąr
ps ´ r ´ 1q!
ż
ż
1
1 w r 1 w 3 ´ 1 z 3 ´ξw `τ z
3
`
e3
dz
dw
,
p2πiq2 LL `Cout
w ´ z zs
LR `Cin
where the contours are defined in the figure below, and 1aăb is the
indicator function for a ă b.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Definition of the Cusp-Airy Kernel
LL
π
3
LR
Cout
y
Cin
0
x
Figure: Integration contours for the Cusp-Airy kernel.
The Cusp-Airy Process
π
3
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Basic Notation
pnq
pnq
pnq
pnq
pnq
Let tan un , tbn un , ttc un , tt1 un , tt2 un , txc un , and tyc un be suitably
chosen sequences convergent to a, b, tc , t1 , t2 , χc and ηc respectively.
Consider the signed measure
dνpxq “ pχra,t2 s pxq ` χrχc ,bspxq qϕpxqdx ´ p1 ´ ϕpxqqχrt1 ,χc s pxqdx,
where ϕ is the density of µ. Then at the cusp χc ` ηc ´ 1 “ tc , the
asymptotic function can be written as
ż
f pw ; χc , ηc q “ f pw ; χc q “ logpw ´ xqdνpxq.
R
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Assumption 1
Assume that
µn :“
n
1ÿ
δ pnq á µ
n i“1 xi {n
as n Ñ 8, in the sense of weak convergence of measures, where µ is a
positive Borel measure on R.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Assumption 2
Let tc P R2 and let pχc , ηc q :“ pχE ptc q, ηE ptc qq. Assume that
f 1 ptc ; χc , ηc q “ f 2 ptc ; χc , ηc q “ 0 so that pχc , ηc q is the asymptotic cusp
point.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Assumption 3
Assume that for every ε ą 0 and n large enough we have for tc P R2 ,
ˇ
µn ˇrt
2 `ε,t1 ´εs
“
1
n
ÿ
δk{n .
tntc uďkďnpt1 ´εq
where txu “ maxtm P Z : m ď xu.
The Cusp-Airy Process
(1)
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Assumption 3
Assume that there is a neighbourhood U of tc such that
lim n2{3 pfn1 pzq ´ f 1 pzqq “ 0
nÑ8
uniformly in U.
The Cusp-Airy Process
(2)
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Rescaled Coordinate System
Fix r , s P Z and define the rescaled variables ξn , τn P R, by
$
`
˘
pnq
1
1{3
’
’ x1 “ nxc ` 2 `r ´ c0 n ξn ˘
’
&
pnq
y1 “ nyc ` 21 r ` c0 n1{3 ξn
`
˘ ,
pnq
’
x2 “ nxc ` 21 s ´ c0 n1{3 τn
’
’
`
˘
%
pnq
y2 “ nyc ` 21 s ` c0 n1{3 τn
where c0 is a suitable constant. We assume that
lim ξn “ ξ,
nÑ8
lim τn “ τ.
nÑ8
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Rescaled Coordinate System
pnq
nxc
pnq
` nyc
pnq
“ ntc
`n
r
pnq
pnq
pnxc , nyc q
ξ
Figure: Rescaled coordinate system at the cusp.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Theorem
Under our assumptions and the scaling above the following result holds:
lim
nÑ8
pn px2 , y2 q c0 1{3 pnq
n KR ppx1 , y1 q, px2 , y2 qq “ KCA ppξ, r q, pτ, sqq
pn px1 , y1 q 2
uniformly for ξ and τ in some fixed compact subset of R, and where
pn px2 , y2 q
pn px1 , y1 q
is a suitable conjugation factor.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Theorem
Let pξjr , r q be the rescaled coordinates for particles on line r . Fix
r1 , . . . , rM P Z and let φ : R ˆ tr1 , . . . , rM u Ñ r0, 1s be a bounded
measurable function with compact support. Let Ex pnq denote the
expectation with respect to the determinantal point process with kernel
pnq
KR . Then,
„

ź
ź
lim Ex pnq
p1 ´ φpξjr , r qq “ detpI ´ φKCA qL2 pRˆtr1 ,...,rM uq .
nÑ8
r Ptr1 ,...,rm u j
(3)
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Cusp-Airy Process in Random Skew Plane Partition Models
This type of cusp situation in a random lozenge tiling model was also
discovered and discussed briefly by Okounkov and Reshetikhin in [8] who
called it a Cuspidal turning point. However, the integration contours in
their formula are not correct. Also, no proof is given in [8].
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Non-exact Discrete Cancellation
We note that asymptotically, we have a perfect cancellation between µ
and λ on the interval rtc , t2 s. For finite n, this cancellation is not exact,
and will depend on our fixed discrete parameters r and s. More precisely,
with
ntcpnq `r ´1
qn pw ; r q “ 1r ą0
ź
ntcpnq ´1
ź
pw ´ kq ` 1r “0 ` 1r ă0
pnq
k“ntc
pnq
k“ntc `r
we can write the integrand as
qn pw ; r q e npgn,1 pw q´npgn,2 pzqq
qn pz; sq
w ´z
The Cusp-Airy Process
pw ´ kq´1 ,
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Changing The Integration Contours
To preform a steepest descent analysis of e npgn,1 pw q´npgn,2 pzqq , we must
change the integration contours. This is done by using the residue
theorem.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Global descent and ascent contours
We show that away from the critical point tc , we can chose the descent
and ascent contours of gn,1 and gn,2 to be those of the the asymptotic
function f pw ; χc q. The existence of these contours are proven abstractly
using the structure of the support of the signed measure µ ´ λ|ra,bs and
properties of harmonic functions.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Local descent and ascent contours
The local contours around critical point tc of gn,1 and gn,2 are glued
together with those of f pw ; χc q outside some ball of fixed radius . The
local analysis inside the ball is done using Taylor expansion and the
convergence speed assumption of Assumption 3.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Reflection Symmetry
The Cusp-Airy kernel satisfies
KCA ppξ, ´r q, pτ, ´sqq “ p´1qs´r KCA ppτ, sq, pξ, r qq.
In particular, this implies that the correlation functions satisfies the
reflection symmetry
ρn ppξ1 , ´r1 q, pξ2 , ´r2 q, ..., pξn , ´rn qq “ ρn ppξ1 , r1 q, pξ2 , r2 q, ..., pξn , rn qq
for all n.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Representation of The Cusp-Airy Kernel
In this section we give an alternative representation of the Cusp-Airy
kernel involving the so called r-Airy integrals and certain polynomials.
Define the r-Airy integrals,
A˘
r puq “
1
2π
ż
1
e 3 ia
3
`iua
p¯iaq˘r da,
`
where r ě 0 and ` is a contour from 8e 5πi{6 to 8e πi{6 such that 0 lies
above the contour, see [1]; compare also with the functions s pmq and t pmq
in [3]. Note that A˘
0 puq “ Ai puq, the standard Airy function.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Representation of The Cusp-Airy Kernel
Define the polynomials Pn pw , ξq and pn pξq through
1
Pn pw , ξq :“ e ´ 3 w
3
`uw
d n 1 w 3 ´uw
e3
dw n
and
pn puq :“ Pn p0, uq.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Representation of The Cusp-Airy Kernel
We can now give a different formula for the Cusp-Airy kernel in terms of
the r-Airy integrals.
KCA ppξ, r q, pτ, sqq “ ´1τ ěξ 1sąr
pτ ´ ξqs´r ´1
` K̃CA ppξ, r q, pτ, sqq,
ps ´ r ´ 1q!
where K̃CA is given
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Representation of The Cusp-Airy Kernel
(i) for r , s ě 0, by
ż8
K̃CA ppξ, r q, pτ, sqq “
`
A´
s pτ ` λqAr pξ ` λq dλ,
0
(ii) for r ě 0, s ă 0, by
ż8
K̃CA ppξ, r q, pτ, sqq “ p´1qs
`
A`
´s pτ ` λqAr pξ ` λq dλ,
0
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Representation of The Cusp-Airy Kernel
(iii) for r ă 0, s ě 0, by
ż8
´
K̃CA ppξ, r q, pτ, sqq “ p´1qr
A´
s pτ ` λqA´r pξ ` λq dλ
0
ż8
s´r
` p´1q
ps´1 pτ ` λqA´
´r pξ ` λq dλ
` p´1qs´r
0
s´1
ÿ
pk pτ qA´r `s´k pξq `
s´r
ÿ´1
k“0
p´1qk pk pτ qps´r ´1´k pξq,
k“0
(iv) and for r , s ă 0, by
ż8
K̃CA ppξ, r q, pτ, sqq “ p´1qs´r
´
A`
´s pτ ` λqA´r pξ ` λq dλ.
0
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
Similar Processes
§
The GUE corner process.
§
The “The Cusp-Airy version of the Tacnode process”. Uses
skew-Young Tableaux. Can also be thought of a double-sided version
of the GUE corner process with an additional overlap parameter r . Is
studied in [2].
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
[1] M. Adler, J. Delpine and P. van Moerbecke,
Dyson’s Nonintersecting Brownian Motions with a Few Outliers.
Communications on Pure and Applied Mathematics 62 (2009), no. 3,
334–395
[2] M. Adler, K. Johansson and P. van Moerbecke,
Tilings of non-convex Polygons, skew-Young Tableaux and
determinantal Processes. arXiv:1609.06995
[3] J. Baik, G. Ben Arous and S. Péché,
Phase transition of the largest eigenvalue for non-null complex sample
covariance matrices. Ann. Probab. 33 (2005), no. 5, 1643–1697
[4] E. Duse and A. Metcalfe.
Asymptotic geometry of discrete interlaced patterns: Part I.
International Journal of Mathematics. 26, 11, (2015) 1550093 (66
pages).
[5] E. Duse and A. Metcalfe.
Asymptotic geometry of discrete interlaced patterns: Part II.
arXiv:1507.00467.
The Cusp-Airy Process
Our Model: Uniformly Random Discrete Gelfand-Tsetlin Patterns
The Cusp-Airy Process
A Few Proof Ideas
Properties of The Cusp-Airy Kernel
Similar Processes
[6] E. Duse and A. Metcalfe.
Universal edge fluctuations of discrete interlaced particle systems. (In
preparation. Estimated 2017).
[7] E. Duse and K. Johansson and A. Metcalfe.
Cusp Airy Process . Electron. J. Prob. Volume 21 (2016), paper no.
57, 50 pp.
[8] A. Okounkov and N. Reshetikhin,
The birth of a random matrix. Mosc. Math. J. 6 (2006), no. 3,
553–566
[9] L. Petrov,
Asymptotics of Random Lozenge Tilings via Gelfand-Tsetlin Schemes.
Probab. Theory Related Fields 160 (2014), no. 3-4, 429–487
The Cusp-Airy Process