Integer partitions and exclusion statistics
Limit shapes and the largest part of Young diagrams
Sanjib Sabhapandit
Laboratoire de Physique Théorique et Modèles Statistiques
CNRS UMR 8626 — Université Paris-Sud
91405 Orsay cedex, France
Collaborators
Alain Comtet
Satya N. Majumdar
Stéphane Ouvry
References
1. J. Stat. Mech. (2007) P10001
2. J. Math. Phys. Anal. Geom. 4, 1 (2007)
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Exclusion statistics
Integer partitions
Integer partitions
ideal bosons/fermions
Minimal difference partitions
exclusion statistics
On minimal difference partitions
1
Limit shapes of Young diagrams
1
2
3
4
2
Largest part of Young diagrams
1
2
3
3
Earlier results
Our results
Derivation
Physical interpretation
Our result & earlier result
Derivation
Restricted grand partition function
Summary
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
1 / 22
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Exclusion statistics
Integer partitions
Integer partitions
ideal bosons/fermions
Minimal difference partitions
exclusion statistics
On minimal difference partitions
1
Limit shapes of Young diagrams
1
2
3
4
2
Largest part of Young diagrams
1
2
3
3
Earlier results
Our results
Derivation
Physical interpretation
Our result & earlier result
Derivation
Restricted grand partition function
Summary
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
1 / 22
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Exclusion statistics
Integer partitions
Integer partitions
ideal bosons/fermions
Minimal difference partitions
exclusion statistics
On minimal difference partitions
1
Limit shapes of Young diagrams
1
2
3
4
2
Largest part of Young diagrams
1
2
3
3
Earlier results
Our results
Derivation
Physical interpretation
Our result & earlier result
Derivation
Restricted grand partition function
Summary
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
1 / 22
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Exclusion statistics
Integer partitions
Integer partitions
ideal bosons/fermions
Minimal difference partitions
exclusion statistics
On minimal difference partitions
1
Limit shapes of Young diagrams
1
2
3
4
2
Largest part of Young diagrams
1
2
3
3
Earlier results
Our results
Derivation
Physical interpretation
Our result & earlier result
Derivation
Restricted grand partition function
Summary
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
1 / 22
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Exclusion statistics
Integer partitions
Integer partitions
ideal bosons/fermions
Minimal difference partitions
exclusion statistics
On minimal difference partitions
1
Limit shapes of Young diagrams
1
2
3
4
2
Largest part of Young diagrams
1
2
3
3
Earlier results
Our results
Derivation
Physical interpretation
Our result & earlier result
Derivation
Restricted grand partition function
Summary
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
1 / 22
Statistics of n identical particles in m degenerate states
Bose statistics
Fermi statistics
#{particles in each state}= 0, 1, 2, . . .
#{particles in each state}= 0 or 1.
•
|• •
•••• ••
{z• • • • • • • • •}
wB =
Z=
Choosing n out of m boxes
n particles + (m−1) partitions
∞
X
(n + m − 1)!
wF =
n! (m − 1)!
wB z n e−nβ = 1 − z e−β
−m
Z=
m
X
m!
n! (m − n)!
wF z n e−nβ = 1 + z e−β
m
n=0
n=0
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
2 / 22
Statistics of n identical particles in m degenerate states
Bose statistics
Fermi statistics
#{particles in each state}= 0, 1, 2, . . .
#{particles in each state}= 0 or 1.
•
|• •
•••• ••
{z• • • • • • • • •}
wB =
Z=
Choosing n out of m boxes
n particles + (m−1) partitions
∞
X
(n + m − 1)!
m → m − (n − 1)
wF =
n! (m − 1)!
wB z n e−nβ = 1 − z e−β
−m
Z=
m
X
m!
n! (m − n)!
wF z n e−nβ = 1 + z e−β
m
n=0
n=0
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
2 / 22
Statistics of n identical particles in m degenerate states
Bose statistics
Fermi statistics
#{particles in each state}= 0, 1, 2, . . .
#{particles in each state}= 0 or 1.
•
|• •
•••• ••
{z• • • • • • • • •}
(m → m + ∆m)
wB =
Z=
Choosing n out of m boxes
n particles + (m−1) partitions
∞
X
m → m − (n − 1)
(n + m − 1)!
wF =
n! (m − 1)!
wB z n e−nβ = 1 − z e−β
−m
Z=
m
X
m!
n! (m − n)!
wF z n e−nβ = 1 + z e−β
m
n=0
n=0
Exclusion statistics: interpolation between Bose and Fermi statistics
[Haldane (1991)] :
wE =
[n + m − p(n − 1) − 1]!
n! [m − p(n − 1) − 1]!
∆m = −p∆n
(∆n = n − 1)
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
2 / 22
Statistics of n identical particles in m degenerate states
Bose statistics
Fermi statistics
#{particles in each state}= 0, 1, 2, . . .
#{particles in each state}= 0 or 1.
•
|• •
•••• ••
{z• • • • • • • • •}
(m → m + ∆m)
wB =
Z=
Choosing n out of m boxes
n particles + (m−1) partitions
∞
X
m → m − (n − 1)
(n + m − 1)!
wF =
n! (m − 1)!
wB z n e−nβ = 1 − z e−β
−m
Z=
m
X
m!
n! (m − n)!
wF z n e−nβ = 1 + z e−β
m
n=0
n=0
Exclusion statistics: interpolation between Bose and Fermi statistics
[Haldane (1991)] :
wE =
[n + m − p(n − 1) − 1]!
p = 0 ⇒ wE = wB
n! [m − p(n − 1) − 1]!
p = 1 ⇒ wE = wF
∆m = −p∆n
(∆n = n − 1)
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
2 / 22
Statistics of n identical particles in m degenerate states
Bose statistics
Fermi statistics
#{particles in each state}= 0, 1, 2, . . .
#{particles in each state}= 0 or 1.
•
|• •
•••• ••
{z• • • • • • • • •}
wB =
Z=
Choosing n out of m boxes
n particles + (m−1) partitions
∞
X
(n + m − 1)!
wF =
n! (m − 1)!
wB z n e−nβ = 1 − z e−β
−m
Z=
m
X
m!
n! (m − n)!
wF z n e−nβ = 1 + z e−β
m
n=0
n=0
Exclusion statistics: interpolation between Bose and Fermi statistics
[Haldane (1991)] :
wE =
[Polychronakos (1996)] :
wE =
[n + m − p(n − 1) − 1]!
p = 0 ⇒ wE = wB
n! [m − p(n − 1) − 1]!
p = 1 ⇒ wE = wF
m [n + m − pn − 1]!
n! [m − pn]!
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
2 / 22
Statistics of n identical particles in m degenerate states
Bose statistics
Fermi statistics
#{particles in each state}= 0, 1, 2, . . .
#{particles in each state}= 0 or 1.
•
|• •
•••• ••
{z• • • • • • • • •}
wB =
Z=
Choosing n out of m boxes
n particles + (m−1) partitions
∞
X
(n + m − 1)!
wF =
n! (m − 1)!
wB z n e−nβ = 1 − z e−β
−m
Z=
m
X
m!
n! (m − n)!
wF z n e−nβ = 1 + z e−β
m
n=0
n=0
Exclusion statistics: interpolation between Bose and Fermi statistics
[Haldane (1991)] :
wE =
[Polychronakos (1996)] :
wE =
[n + m − p(n − 1) − 1]!
p = 0 ⇒ wE = wB
n! [m − p(n − 1) − 1]!
p = 1 ⇒ wE = wF
m [n + m − pn − 1]!
n! [m − pn]!
wE (m, n) = wE (m − 1, n) + wE (m − p , n − 1)
S. Sabhapandit (LPTMS, Orsay, France)
[for integer p]
Limit shapes and largest part of Young diagrams
2 / 22
Statistics of n identical particles in m degenerate states
Bose statistics
Fermi statistics
#{particles in each state}= 0, 1, 2, . . .
#{particles in each state}= 0 or 1.
•
|• •
•••• ••
{z• • • • • • • • •}
wB =
Z=
Choosing n out of m boxes
n particles + (m−1) partitions
∞
X
(n + m − 1)!
wF =
n! (m − 1)!
wB z n e−nβ = 1 − z e−β
−m
Z=
m
X
m!
n! (m − n)!
wF z n e−nβ = 1 + z e−β
m
n=0
n=0
Exclusion statistics: interpolation between Bose and Fermi statistics
[Haldane (1991)] :
wE =
[Polychronakos (1996)] :
wE =
[n + m − p(n − 1) − 1]!
p = 0 ⇒ wE = wB
n! [m − p(n − 1) − 1]!
p = 1 ⇒ wE = wF
m [n + m − pn − 1]!
n! [m − pn]!
m
Z = yp z e−β
yp (x ) − x yp1−p (x ) = 1
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
2 / 22
Statistics of n identical particles in m degenerate states
Bose statistics
Fermi statistics
#{particles in each state}= 0, 1, 2, . . .
#{particles in each state}= 0 or 1.
•
|• •
•••• ••
{z• • • • • • • • •}
wB =
Z=
Choosing n out of m boxes
n particles + (m−1) partitions
∞
X
(n + m − 1)!
wF =
n! (m − 1)!
wB z n e−nβ = 1 − z e−β
−m
Z=
m
X
m!
n! (m − n)!
wF z n e−nβ = 1 + z e−β
m
n=0
n=0
Exclusion statistics: interpolation between Bose and Fermi statistics
[Haldane (1991)] :
wE =
[Polychronakos (1996)] :
wE =
y0 (x ) = (1 − x )
−1
,
[n + m − p(n − 1) − 1]!
p = 0 ⇒ wE = wB
n! [m − p(n − 1) − 1]!
p = 1 ⇒ wE = wF
m [n + m − pn − 1]!
n! [m − pn]!
y1 (x ) = 1 + x
S. Sabhapandit (LPTMS, Orsay, France)
p = 0, 1
m
Z = yp z e−β
yp (x ) − x yp1−p (x ) = 1
Limit shapes and largest part of Young diagrams
2 / 22
Statistics of n identical particles in m degenerate states
Bose statistics
Fermi statistics
#{particles in each state}= 0, 1, 2, . . .
#{particles in each state}= 0 or 1.
•••• ••
{z• • • • • • • • •}
∞
X
(n + m − 1)!
wF =
n! (m − 1)!
wB z n e−nβ = 1 − z e−β
−m
Z=
m
X
m!
wB =
Z=
Choosing n out of m boxes
n particles + (m−1) partitions
n! (m − n)!
wF z n e−nβ = 1 + z e−β
ln Z = m ln yp z e−β
•
|• •
m
n=0
n=0
Exclusion statistics: interpolation between Bose and Fermi statistics
[Haldane (1991)] :
wE =
[Polychronakos (1996)] :
wE =
y0 (x ) = (1 − x )
−1
,
[n + m − p(n − 1) − 1]!
p = 0 ⇒ wE = wB
n! [m − p(n − 1) − 1]!
p = 1 ⇒ wE = wF
m [n + m − pn − 1]!
n! [m − pn]!
y1 (x ) = 1 + x
S. Sabhapandit (LPTMS, Orsay, France)
p = 0, 1
m
Z = yp z e−β
yp (x ) − x yp1−p (x ) = 1
Limit shapes and largest part of Young diagrams
2 / 22
Exclusion statistics (thermodynamics)
If the grand partition function of a quantum gas
Z (β, z ) =
∞
X
zN
N=0
X
e−β E ρ(E , N)
E
| {z }
micro-canonical partition function
can be expressed as an integral representation
Z
A
ln Z (β, z ) =
0
∞
ρ̃() ln yp z e−β d,
|{z}
single particle density of states
where
B
yp (x ) − x yp1−p (x ) = 1,
then the gas is said to obey exclusion statistics with parameter 0 ≤ p ≤ 1.
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
3 / 22
Exclusion statistics (thermodynamics)
If the grand partition function of a quantum gas
Z (β, z ) =
∞
X
zN
N=0
X
e−β E ρ(E , N)
| {z }
E
micro-canonical partition function
can be expressed as an integral representation
Z
A
ln Z (β, z ) =
0
∞
ρ̃() ln yp z e−β d,
|{z}
single particle density of states
where
B
yp (x ) − x yp1−p (x ) = 1,
then the gas is said to obey exclusion statistics with parameter 0 ≤ p ≤ 1.
1
p=0
y0 (x ) =
2
p=1
y1 (x ) = 1 + x
3
0<p<1
1
1−x
Bose statistics.
Fermi statistics.
fractional exclusion statistics.
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
3 / 22
Microscopic models
Z
ln Z (β, z ) =
∞
ρ̃() ln yp z e−β d
where
yp (x ) − x yp1−p (x ) = 1.
0
1
Anyon gas in a strong magnetic field
eφ
[de Veigy & Ouvry (1994)]
eφ

2
N
X
X k × rij
eB
1
pi − α
−
k × ri 
H=
2
i=1
2m
j 6=i
rij
2
eφ
eφ
eφ
eB
α = φ/φ0 ≡ −p ∈ [−1, 0], φ0 = 2π/e , rij = ri − rj
ρ̃() =
VB
φ0
δ ( − ωc )
ωc =
eB
2m
S. Sabhapandit (LPTMS, Orsay, France)
=
1
2
× (cyclotron frequency)
Limit shapes and largest part of Young diagrams
4 / 22
Microscopic models
Z
∞
ln Z (β, z ) =
ρ̃() ln yp z e−β d
where
yp (x ) − x yp1−p (x ) = 1.
0
1
Anyon gas in a strong magnetic field
eφ
[de Veigy & Ouvry (1994)]
eφ

2
N
X
X k × rij
eB
1
pi − α
−
k × ri 
H=
2
i=1
2m
rij
j 6=i
eφ
eφ
eφ
eB
2
α = φ/φ0 ≡ −p ∈ [−1, 0], φ0 = 2π/e , rij = ri − rj
ρ̃() =
2
VB
φ0
δ ( − ωc )
ωc =
eB
2m
=
One-dimensional Calogero model
H=−
N
1 X ∂2
2
i=1
∂ xi2
(1) ω → 0 case:
+
1
2
× (cyclotron frequency)
[Murthy & Shankar (1994), Isakov (1994)]
N
X α(1 + α)
1 2 X 2
+
ω
xi ,
(xi − xj )2
2
i <j
i=1
ρ̃() =
1
ω
S. Sabhapandit (LPTMS, Orsay, France)
α ≡ −p ∈ [−1, 0].
(2) ω = 0 case:
ρ̃() =
L
√
π 2
Limit shapes and largest part of Young diagrams
4 / 22
Bose/Fermi statistics
Integer partitions
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
5 / 22
Integer partitions
A partition of a positive integer E is a decomposition of E as a sum of a
nonincreasing sequence of positive integers {hj }, i.e.,
E =
X
hj
such that
hj ≥ hj+1 ,
for
j = 1, 2 . . . .
j
Example
4=4
=3+1
=2+2
4
3+1
=2+1+1
=1+1+1+1
2+2
Young diagrams
(Ferrers diagrams)
2+1+1
1+1+1+1
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
6 / 22
Integer partitions
A partition of a positive integer E is a decomposition of E as a sum of a
nonincreasing sequence of positive integers {hj }, i.e.,
E =
X
hj
such that
hj ≥ hj+1 ,
for
j = 1, 2 . . . .
j
Example
ρ(E ) := #{partitions of E }.
ρ(4) = 5
ρ(5) = 7
4=4
=3+1
=2+2
4
3+1
=2+1+1
..
.
=1+1+1+1
2+2
Young diagrams
(Ferrers diagrams)
ρ(10) = 42
ρ(100) = 190569292
2+1+1
1+1+1+1
S. Sabhapandit (LPTMS, Orsay, France)
ρ(E ) ≈
1
4E
√ eπ
√
2E /3
3
[Hardy & Ramanujan (1918)]
Limit shapes and largest part of Young diagrams
6 / 22
Integer partitions
A partition of a positive integer E is a decomposition of E as a sum of a
nonincreasing sequence of positive integers {hj }, i.e.,
E =
X
hj
such that
hj ≥ hj+1 ,
for
j = 1, 2 . . . .
j
Example
ρ(E ) := #{partitions of E }.
ρ(4) = 5
ρ(5) = 7
4=4
=3+1
=2+2
4
3+1
=2+1+1
ρ(10) = 42
ρ(100) = 190569292
..
.
=1+1+1+1
2+2
Young diagrams
(Ferrers diagrams)
2+1+1
1+1+1+1
ρ(E ) ≈
1
4E
√ eπ
√
2E /3
3
[Hardy & Ramanujan (1918)]
Partitions into distinct parts
ρ(4) = 2
E =
X
hj
such that hj > hj+1 .
j
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
6 / 22
Integer partitions
A partition of a positive integer E is a decomposition of E as a sum of a
nonincreasing sequence of positive integers {hj }, i.e.,
E =
X
hj
such that
hj ≥ hj+1 ,
for
j = 1, 2 . . . .
j
Example
ρ(E ) := #{partitions of E }.
ρ(4) = 5
ρ(5) = 7
4=4
=3+1
=2+2
4
3+1
=2+1+1
ρ(10) = 42
ρ(100) = 190569292
..
.
=1+1+1+1
2+2
Young diagrams
(Ferrers diagrams)
2+1+1
1+1+1+1
ρ(E ) ≈
1
4E
√ eπ
√
2E /3
3
[Hardy & Ramanujan (1918)]
Partitions into distinct parts
E =
X
hj
such that hj > hj+1 .
j
ρ(4) = 2
ρ(100) = 444793
..
. 1
ρ(E ) ≈
4
·
1
31/4 E 3/4
eπ
√
E /3
see [Abramowitz & Stegun (1972)]
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
6 / 22
Integer partitions
ideal bosons/fermions
18
ni := #{columns whose heights = i }.
16
E =
13 13
X
j
hj
hj =
∞
X
ni i
Number of parts N =
9
with i = i .
i=1
∞
X
ni .
i=1
6
5
5
3 3
j
91 = 18 + 16 + 13 + 13 + 9
+6+5+5+3+3
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
7 / 22
Integer partitions
ideal bosons/fermions
18
ni := #{columns whose heights = i }.
16
E =
13 13
X
hj =
j
hj
∞
X
ni i
Number of parts N =
9
with i = i .
i=1
∞
X
ni .
i=1
6
1
5
hj ≥ hj+1
ni = 0, 1, . . . , ∞ (bosons).
5
3 3
j
91 = 18 + 16 + 13 + 13 + 9
+6+5+5+3+3
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
7 / 22
Integer partitions
ideal bosons/fermions
18
ni := #{columns whose heights = i }.
16
E =
13 13
X
hj =
j
hj
∞
X
ni i
Number of parts N =
9
with i = i .
i=1
∞
X
ni .
i=1
6
5
1
hj ≥ hj+1
ni = 0, 1, . . . , ∞ (bosons).
2
hj > hj+1
ni = 0, 1 (fermions).
5
3 3
j
91 = 18 + 16 + 13 + 13 + 9
+6+5+5+3+3
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
7 / 22
Integer partitions
ideal bosons/fermions
18
ni := #{columns whose heights = i }.
16
E =
13 13
X
hj =
ni i
Number of parts N =
9
with i = i .
i=1
j
hj
∞
X
∞
X
ni .
i=1
6
5
1
hj ≥ hj+1
ni = 0, 1, . . . , ∞ (bosons).
2
hj > hj+1
ni = 0, 1 (fermions).
5
3 3
Number of ways of partitioning E
j
91 = 18 + 16 + 13 + 13 + 9
z }| {
ρ(E )
+6+5+5+3+3
S. Sabhapandit (LPTMS, Orsay, France)
=
X
{ni }
δ
E−
∞
X
!
ni i
.
i=1
Limit shapes and largest part of Young diagrams
7 / 22
Integer partitions
ideal bosons/fermions
18
ni := #{columns whose heights = i }.
16
E =
13 13
X
hj =
ni i
Number of parts N =
9
with i = i .
i=1
j
hj
∞
X
∞
X
ni .
i=1
6
5
1
hj ≥ hj+1
ni = 0, 1, . . . , ∞ (bosons).
2
hj > hj+1
ni = 0, 1 (fermions).
5
3 3
Number of ways of partitioning E into N parts
j
91 = 18 + 16 + 13 + 13 + 9
∞
z }| { X
X
ρ(E , N) =
δ E−
ni i
+6+5+5+3+3
S. Sabhapandit (LPTMS, Orsay, France)
{ni }
i=1
!
δ
N−
∞
X
!
ni
.
i=1
Limit shapes and largest part of Young diagrams
7 / 22
Integer partitions
ideal bosons/fermions
18
ni := #{columns whose heights = i }.
16
E =
13 13
X
hj =
ni i
Number of parts N =
9
with i = i .
i=1
j
hj
∞
X
∞
X
ni .
i=1
6
5
1
hj ≥ hj+1
ni = 0, 1, . . . , ∞ (bosons).
2
hj > hj+1
ni = 0, 1 (fermions).
5
3 3
Number of ways of partitioning E into N parts
j
∞
z }| { X
X
ρ(E , N) =
δ E−
ni i
91 = 18 + 16 + 13 + 13 + 9
{ni }
+6+5+5+3+3
The grand partition function:
Z (β, z ) =
∞
X
N=0
zN
X
e−β E ρ(E , N) =
E
!
δ
N−
i=1
∞
X
!
ni
.
i=1
∞
Y
−1



1 − z e−β i


 i=1
(bosons)
∞

Y




1 + z e−β i

(fermions)
i=1
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
7 / 22
Exclusion statistics
Minimal difference partitions
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
8 / 22
Minimal difference p partitions
E =
X
hj such that hj − hj+1 ≥ p ,
p = 0, 1, 2, . . .
j
hj
≥p
≥p
≥p
≥p
≥1
j
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
9 / 22
Minimal difference p partitions
E =
X
hj such that hj − hj+1 ≥ p ,
p = 0, 1, 2, . . .
j
Number of ways of partitioning E into N parts
∞ z }| {
X
ρ(E , N) x E =
hj
E =1
x N+pN(N −1)/2
(1 − x )(1 − x 2 ) · · · (1 − x N )
≥p
≥p
≥p
≥p
≥1
j
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
9 / 22
Minimal difference p partitions
E =
X
hj such that hj − hj+1 ≥ p ,
p = 0, 1, 2, . . .
j
Number of ways of partitioning E into N parts
∞ z }| {
X
ρ(E , N) x E =
hj
E =1
x N+pN(N −1)/2
(1 − x )(1 − x 2 ) · · · (1 − x N )
x = e−β
≥p
∞
X
≥p
Grand partition function
N=0
zN
z
X
}|
{
e−β E ρ(E , N)
E
z }| {
Z (β, z )
≥p
≥p
≥1
j
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
9 / 22
Minimal difference p partitions
E =
X
hj such that hj − hj+1 ≥ p ,
p = 0, 1, 2, . . .
j
Number of ways of partitioning E into N parts
∞ z }| {
X
ρ(E , N) x E =
hj
E =1
x N+pN(N −1)/2
(1 − x )(1 − x 2 ) · · · (1 − x N )
x = e−β
≥p
∞
X
≥p
Grand partition function
N=0
zN
z
X
{
E
z }| { β→0 Z
ln Z (β, z ) −−−−−→
≥p
}|
e−β E ρ(E , N)
∞
ln yp z e−β d
1
≥p
where
≥1
yp (x ) − x yp1−p (x ) = 1.
Derivation
[Comtet, Majumdar & Ouvry (2007)]
j
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
9 / 22
Minimal difference p partitions
E =
X
hj such that hj − hj+1 ≥ p ,
p = 0, 1, 2, . . .
j
Number of ways of partitioning E into N parts
∞ z }| {
X
ρ(E , N) x E =
hj
E =1
x N+pN(N −1)/2
(1 − x )(1 − x 2 ) · · · (1 − x N )
x = e−β
≥p
∞
X
≥p
Grand partition function
N=0
zN
z
X
{
E
z }| { β→0 Z
ln Z (β, z ) −−−−−→
≥p
}|
e−β E ρ(E , N)
∞
ln yp z e−β d
1
≥p
where
≥1
j
yp (x ) − x yp1−p (x ) = 1.
∞
X
n=1
bn z n
[Comtet, Majumdar & Ouvry (2007)]
ln yp (x ) =
∞
n −1
X
xn Y h
n=1
S. Sabhapandit (LPTMS, Orsay, France)
Derivation
n
1−
k=1
Limit shapes and largest part of Young diagrams
pn i
k
9 / 22
Minimal difference p partitions
E =
X
hj such that hj − hj+1 ≥ p ,
p = 0, 1, 2, . . .
j
Number of ways of partitioning E into N parts
∞ z }| {
X
ρ(E , N) x E =
hj
E =1
x N+pN(N −1)/2
(1 − x )(1 − x 2 ) · · · (1 − x N )
x = e−β
≥p
∞
X
≥p
Grand partition function
N=0
zN
z
X
{
E
z }| { β→0 Z
ln Z (β, z ) −−−−−→
≥p
}|
e−β E ρ(E , N)
∞
ln yp z e−β d
1
≥p
where
≥1
j
yp (x ) − x yp1−p (x ) = 1.
∞
X
n=1
bn z n
Derivation
[Comtet, Majumdar & Ouvry (2007)]
ln yp (x ) =
∞
n −1
X
xn Y h
n=1
n
1−
k=1
pn i
k
When one analytically continues the results to noninteger values of p,
for 0 < p < 1, the minimal difference p partition corresponds to
a gas of quantum particles obeying fractional exclusion statistics.
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
9 / 22
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Exclusion statistics
Integer partitions
Integer partitions
ideal bosons/fermions
Minimal difference partitions
exclusion statistics
On minimal difference partitions
1
Limit shapes of Young diagrams
1
2
3
4
2
Largest part of Young diagrams
1
2
3
3
Earlier results
Our results
Derivation
Physical interpretation
Our result & earlier result
Derivation
Restricted grand partition function
Summary
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
1 / 22
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Exclusion statistics
Integer partitions
Integer partitions
ideal bosons/fermions
Minimal difference partitions
exclusion statistics
On minimal difference partitions
1
Limit shapes of Young diagrams
1
2
3
4
2
Largest part of Young diagrams
1
2
3
3
Earlier results
Our results
Derivation
Physical interpretation
Our result & earlier result
Derivation
Restricted grand partition function
Summary
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
1 / 22
Limit shapes
hj
≥p
≥p
h
≥p
Wh
≥p
≥1
j
Let
E →∞
Y = lim
E →∞
Wh
√
X = lim
,
E
h
√
E
.
The limit shape is given by
the XY curve.
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
10 / 22
Limit shapes
hj
Earlier results
≥p
p = 0 (unrestricted partitions):
e−b(0)X + e−b(0)Y = 1,
≥p
h
6
[Temperley (1952)]
≥p
Wh
π
b(0) = √
≥p
≥1
j
Let
E →∞
Y = lim
E →∞
Wh
√
X = lim
,
E
h
√
E
.
The limit shape is given by
the XY curve.
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
10 / 22
Limit shapes
hj
Earlier results
≥p
p = 0 (unrestricted partitions):
e−b(0)X + e−b(0)Y = 1,
≥p
h
[Temperley (1952)]
≥p
Wh
≥p
j
Let
E →∞
Y = lim
E →∞
Wh
√
√
E
[Vershik and collaborators]
eb(1)X − e−b(1)Y = 1,
π
b(1) = √
12
[Vershik and collaborators]
,
E
h
6
p = 1 (partitions into distinct parts):
≥1
X = lim
,
π
b(0) = √
.
The limit shape is given by
the XY curve.
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
10 / 22
Limit shapes
hj
Earlier results
≥p
p = 0 (unrestricted partitions):
e−b(0)X + e−b(0)Y = 1,
≥p
h
[Temperley (1952)]
≥p
Wh
≥p
,
π
b(0) = √
[Vershik and collaborators]
p = 1 (partitions into distinct parts):
≥1
eb(1)X − e−b(1)Y = 1,
j
π
b(1) = √
Let
E →∞
Y = lim
E →∞
Wh
√
X = lim
√
E
12
[Vershik and collaborators]
,
E
h
6
p = 2 (minimal difference 2 partitions):
.
eb(2)X =
The limit shape is given by
the XY curve.
S. Sabhapandit (LPTMS, Orsay, France)
1h
2
1+
p
i
1 + 4e−b(2)Y ,
π
b(2) = √
15
[Romik (2003)]
Limit shapes and largest part of Young diagrams
10 / 22
Our (general) formulæ for limit shapes
A
X =
1
b(p)
ln yp e
−b(p)Y
hj
E →∞
≥p
Wh
√
X = lim
Y = lim
E →∞
≥p
h
in which
yp (w ) − w yp1−p (w ) = 1
Wh
Z
b (p) =
≥1
ln yp e
=
6
j
d
0
π2
E
≥p
∞
−
≥p
and
2
E
h
√
π
− Li2 (1/y ∗ ) −
where y ∗ = yp (1)
and Li2 (z ) =
2
(ln y ∗ )2
k2
b(0) = √
6
π
b(1) = √
i.e., y ∗ − y ∗1−p = 1
∞
X
zk
k=1
p
12
π
b(2) = √
15
is the dilogarithm function. b(3) = 0.752617 . . .
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
11 / 22
Our (general) formulæ for limit shapes
A
X =
1
b(p)
ln yp e
−b(p)Y
hj
E →∞
≥p
or
B
Wh
√
X = lim
Y = lim
Y =−
1
b(p)
E →∞
≥p
ln 1 − e−b(p)X − pX
h
in which
yp (w ) − w yp1−p (w ) = 1
Wh
Z
b (p) =
≥1
ln yp e
=
6
j
d
0
π2
E
≥p
∞
−
≥p
and
2
E
h
√
π
− Li2 (1/y ∗ ) −
where y ∗ = yp (1)
and Li2 (z ) =
2
(ln y ∗ )2
k2
b(0) = √
6
π
b(1) = √
i.e., y ∗ − y ∗1−p = 1
∞
X
zk
k=1
p
12
π
b(2) = √
15
is the dilogarithm function. b(3) = 0.752617 . . .
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
11 / 22
Our (general) formulæ for limit shapes
A
X =
1
b(p)
ln yp e
−b(p)Y
b(p)Y
3
E →∞
or
B
Y = lim
Y =−
1
ln 1 − e−b(p)X − pX
b(p)
E →∞
2
Wh
√
X = lim
E
h
√
E
(p = 0)
in which
(p = 1)
1
(p = 2)
yp (w ) − w yp1−p (w ) = 1
(p = 3)
b(p)X
and
b2 (p) =
Z
0
∞
0
ln yp e− d
0
=
π2
6
1
2
3
π
− Li2 (1/y ∗ ) −
where y ∗ = yp (1)
and Li2 (z ) =
2
(ln y ∗ )2
k2
b(0) = √
6
π
b(1) = √
i.e., y ∗ − y ∗1−p = 1
∞
X
zk
k=1
p
12
π
b(2) = √
15
is the dilogarithm function. b(3) = 0.752617 . . .
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
11 / 22
Derivation of the limit shapes
Wh = number of columns
whose heights ≥ h.
hj
≥p
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
h
Wh
≥p
≥p
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
S. Sabhapandit (LPTMS, Orsay, France)
≥1
j
Limit shapes and largest part of Young diagrams
12 / 22
Derivation of the limit shapes
Wh = number of columns
whose heights ≥ h.
hj
≥p
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
h
Wh
≥p
≥p
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
hWh i = z
≥1
j
∂
ln Zh (β, z )
z =1
∂z
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
12 / 22
Derivation of the limit shapes
Wh = number of columns
whose heights ≥ h.
hj
≥p
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
h
Wh
≥p
≥p
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
hWh i = z
≥1
j
∂
ln Zh (β, z )
z =1
∂z
z
}|
{
∂
E =−
ln Z (β, 1)
∂β
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
12 / 22
Derivation of the limit shapes
hj
Wh = number of columns
whose heights ≥ h.
≥p
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
h
Wh
≥p
≥p
Z (β, z ) = Z1 (β, z ) is the full grand
≥1
partition function which counts all the
columns.
A
hWh i = z
j
∂
ln Zh (β, z )
z =1
∂z
z
}|
{
∂
E =−
ln Z (β, 1)
∂β
B
hWh2 i − hWh i2 =
z
∂ ∂
z
ln Zh (β, z )
z =1
∂z ∂z
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
12 / 22
Derivation of the limit shapes
hj
Wh = number of columns
whose heights ≥ h.
≥p
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
h
β→0
ln Zh (β, z ) −
−−→
[yp (w ) − w
≥p
Wh
1
β
yp1−p (w )
Z
∞
βh
= 1]
≥p
ln yp z e− d
≥1
j
Derivation
∂
hWh i = z
ln Zh (β, z )
z =1
∂z
z
}|
{
∂
E =−
ln Z (β, 1)
∂β
B
hWh2 i − hWh i2 =
z
∂ ∂
z
ln Zh (β, z )
z =1
∂z ∂z
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
12 / 22
Derivation of the limit shapes
hj
Wh = number of columns
whose heights ≥ h.
≥p
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
∂
hWh i = z
ln Zh (β, z )
z =1
∂z
h
β→0
ln Zh (β, z ) −
−−→
[yp (w ) − w
≥p
Wh
1
β
yp1−p (w )
Z
∞
≥p
ln yp z e− d
βh
j
Derivation
= 1]
β hWh i = ln yp e−β h
≥1
z
}|
{
∂
E =−
ln Z (β, 1)
∂β
B
hWh2 i − hWh i2 =
z
∂ ∂
z
ln Zh (β, z )
z =1
∂z ∂z
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
12 / 22
Derivation of the limit shapes
hj
Wh = number of columns
whose heights ≥ h.
≥p
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
}|
{
∂
E =−
ln Z (β, 1)
∂β
hWh2 i − hWh i2 =
β
∞
≥p
ln yp z e− d
E
Z
∞
j
Derivation
= 1]
β hWh i = ln yp e−β h
β= √
≥1
βh
yp1−p (w )
b(p)
z
Z
1
β→0
[yp (w ) − w
≥p
Wh
ln Zh (β, z ) −
−−→
∂
hWh i = z
ln Zh (β, z )
z =1
∂z
z
B
h
ln yp e−
1/2
d
0
∂ ∂
z
ln Zh (β, z )
z =1
∂z ∂z
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
12 / 22
Derivation of the limit shapes
hj
Wh = number of columns
whose heights ≥ h.
≥p
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
}|
{
∂
E =−
ln Z (β, 1)
∂β
hWh2 i − hWh i2 =
Z
1
β→0
[yp (w ) − w
≥p
Wh
ln Zh (β, z ) −
−−→
∂
hWh i = z
ln Zh (β, z )
z =1
∂z
z
B
h
β
∞
≥p
ln yp z e− d
βh
yp1−p (w )
β= √
E
Z
∞
j
Derivation
= 1]
β hWh i = ln yp e−β h
b(p)
≥1
ln yp e−
1/2
d
0
"
#
∂ ∂
1 −β h yp0 e−β h
z
z
ln Zh (β, z )
= e
z =1
∂z ∂z
β
yp e−β h
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
12 / 22
Derivation of the limit shapes
hj
Wh = number of columns
whose heights ≥ h.
≥p
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
}|
{
∂
E =−
ln Z (β, 1)
∂β
Z
1
β→0
[yp (w ) − w
≥p
Wh
ln Zh (β, z ) −
−−→
∂
hWh i = z
ln Zh (β, z )
z =1
∂z
z
B
h
β
∞
≥p
ln yp z e− d
βh
yp1−p (w )
β= √
E
Z
∞
j
Derivation
= 1]
β hWh i = ln yp e−β h
b(p)
≥1
ln yp e−
1/2
d
0
"
#
0
−β h
∂
2
2
2 ∂
−β h yp e
β hWh i − hWh i = β z z
ln Zh (β, z )
= βe
z =1
∂z ∂z
yp e−β h
2
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
12 / 22
Derivation of the limit shapes
hj
Wh = number of columns
whose heights ≥ h.
≥p
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
h
[yp (w ) − w
}|
{
∂
E =−
ln Z (β, 1)
∂β
β
∞
≥p
ln yp z e− d
b(p)
β= √
E
Z
∞
≥1
βh
yp1−p (w )
j
Derivation
= 1]
β hWh i = ln yp e−β h
z
B
Z
1
β→0
ln Zh (β, z ) −
−−→
∂
hWh i = z
ln Zh (β, z )
z =1
∂z
≥p
Wh
ln yp e−
1/2
d
0
"
#
0
−β h
∂
2
2
2 ∂
−β h yp e
β hWh i − hWh i = β z z
ln Zh (β, z )
= βe
z =1
∂z ∂z
yp e−β h
2
β→0
β hWh i −−−→ β Wh
S. Sabhapandit (LPTMS, Orsay, France)
Wh
√
E
h E →∞
vs √ −−−→ Limit shape.
E
Limit shapes and largest part of Young diagrams
12 / 22
Derivation of the limit shapes
hj
Wh = number of columns
whose heights ≥ h.
√
X = lim
≥p
Wh
E →∞
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
E →∞
h
[yp (w ) − w
β
∞
b(p)
β= √
≥p
E
∞
E
≥1
βh
yp1−p (w )
Z
h
√
ln yp z e− d
= 1]
ln yp e−
j
Derivation
b(p) X = ln yp e−b(p) Y
z
B
Z
1
β→0
E
≥p
Wh
ln Zh (β, z ) −
−−→
∂
hWh i = z
ln Zh (β, z )
z =1
∂z
}|
{
∂
E =−
ln Z (β, 1)
∂β
Y = lim
1/2
d
0
"
#
0
−β h
∂
2
2
2 ∂
−β h yp e
β hWh i − hWh i = β z z
ln Zh (β, z )
= βe
z =1
∂z ∂z
yp e−β h
2
β→0
β hWh i −−−→ β Wh
S. Sabhapandit (LPTMS, Orsay, France)
Wh
√
E
h E →∞
vs √ −−−→ Limit shape.
E
Limit shapes and largest part of Young diagrams
12 / 22
Physical interpretation of limit shape
hj
β Wh = ln yp e
−β h
where
yp (x )−x
yp1−p (x )
≥p
= 1.
≥p
h
Wh
≥p
≥p
≥1
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
j
13 / 22
Physical interpretation of limit shape
hj
β Wh = ln yp e
−β h
where
yp (x )−x
yp1−p (x )
≥p
= 1.
≥p
Expressing h in terms of Wh yields
h=−
1
β
ln 1 − e
−β Wh
− pWh .
h
Wh
≥p
≥p
≥1
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
j
13 / 22
Physical interpretation of limit shape
j
β Wh = ln yp e−β h
where
yp (x )−x yp1−p (x ) = 1.
Transposed
Young diagram
≥1
≥p
≥p
≥p
Expressing h in terms of Wh yields
h=−
1
β
≥p
Wh
ln 1 − e
−β Wh
− pWh .
S. Sabhapandit (LPTMS, Orsay, France)
hj
h
Limit shapes and largest part of Young diagrams
13 / 22
Physical interpretation of limit shape
j
β Wh = ln yp e−β h
where
yp (x )−x yp1−p (x ) = 1.
Transposed
Young diagram
≥1
≥p
≥p
≥p
Expressing h in terms of Wh yields
h=−
1
β
≥p
Wh
ln 1 − e
−β Wh
− pWh .
hj
h
# particles above energy level Wh
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
13 / 22
Physical interpretation of limit shape
j
β Wh = ln yp e−β h
where
yp (x )−x yp1−p (x ) = 1.
Transposed
Young diagram
≥1
≥p
≥p
≥p
Expressing h in terms of Wh yields
h=−
|
1
β
≥p
Wh
ln 1 − e
{z
−β Wh
bosonic (p = 0)
− pWh .
hj
h
}
# particles above energy level Wh
j
p=0
hj
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
13 / 22
Physical interpretation of limit shape
j
β Wh = ln yp e−β h
where
Transposed
Young diagram
≥1
yp (x )−x yp1−p (x ) = 1.
≥p
≥p
≥p
Expressing h in terms of Wh yields
h=−
|
1
β
≥p
Wh
ln 1 − e
{z
−β Wh
bosonic (p = 0)
hj
− pWh .
h
}
# particles above energy level Wh
j
p=0
p=2
j
hj
S. Sabhapandit (LPTMS, Orsay, France)
hj
Limit shapes and largest part of Young diagrams
13 / 22
Physical interpretation of limit shape
j
β Wh = ln yp e−β h
where
Transposed
Young diagram
≥1
yp (x )−x yp1−p (x ) = 1.
≥p
≥p
≥p
Expressing h in terms of Wh yields
h=−
|
1
β
≥p
Wh
ln 1 − e
{z
−β Wh
bosonic (p = 0)
}
hj
− p Wh .
h
# particles transferred from above Wh to below
# particles above energy level Wh
j
p=0
p=2
j
hj
S. Sabhapandit (LPTMS, Orsay, France)
hj
Limit shapes and largest part of Young diagrams
13 / 22
Physical interpretation of limit shape
j
β Wh = ln yp e−β h
where
Transposed
Young diagram
≥1
yp (x )−x yp1−p (x ) = 1.
≥p
≥p
≥p
Expressing h in terms of Wh yields
h=−
|
1
β
≥p
Wh
ln 1 − e
−β Wh
{z
bosonic (p = 0)
}
hj
− p Wh .
h
# particles transferred from above Wh to below
# particles above energy level Wh
p=0
j
p=2
j
hj
Z
Wh∗
h (Wh ) dWh = E
0
hj
b(p)
β= √
h(Wh∗ ) = 0
S. Sabhapandit (LPTMS, Orsay, France)
E
,
b2 (p) =
π2
6
− Li2 (1/y ∗ ) − p2 (ln y ∗ )2 .
y ∗ − y ∗1−p = 1.
Limit shapes and largest part of Young diagrams
13 / 22
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Exclusion statistics
Integer partitions
Integer partitions
ideal bosons/fermions
Minimal difference partitions
exclusion statistics
On minimal difference partitions
1
Limit shapes of Young diagrams
1
2
3
4
2
Largest part of Young diagrams
1
2
3
3
Earlier results
Our results
Derivation
Physical interpretation
Our result & earlier result
Derivation
Restricted grand partition function
Summary
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
1 / 22
Largest part of Young diagrams: result
minimal difference partitions of E
ρ(E , `) := # such that the largest part h ≤ `
|
{z
} 1
ρ(E , ` → ∞) = ρ(E ) ≡ # partitions
hj
≥p
≥p
≥p
≥p
≥1
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
j
14 / 22
Largest part of Young diagrams: result
minimal difference partitions of E
ρ(E , `) := # such that the largest part h ≤ `
|
{z
} 1
ρ(E , ` → ∞) = ρ(E ) ≡ # partitions
≥p
ρ(E , `)
= Prob[h1 ≤ `]
ρ(E )
S. Sabhapandit (LPTMS, Orsay, France)
≥p
≥p
Consider uniform measure: 1/ρ(E )
C (`|E ) :=
hj
≥p
Limit shapes and largest part of Young diagrams
≥1
j
14 / 22
Largest part of Young diagrams: result
minimal difference partitions of E
ρ(E , `) := # such that the largest part h ≤ `
|
{z
} 1
ρ(E , ` → ∞) = ρ(E ) ≡ # partitions
≥p
≥p
≥p
Consider uniform measure: 1/ρ(E )
C (`|E ) :=
hj
ρ(E , `)
= Prob[h1 ≤ `]
ρ(E )
≥p
≥1
j
We show that (for all p):
E →∞
C (`|E ) −−−−√
−→ F
` E
` − `∗ (E )
σ (E )
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
14 / 22
Largest part of Young diagrams: result
minimal difference partitions of E
ρ(E , `) := # such that the largest part h ≤ `
|
{z
} 1
hj
≥p
ρ(E , ` → ∞) = ρ(E ) ≡ # partitions
≥p
≥p
Consider uniform measure: 1/ρ(E )
C (`|E ) :=
ρ(E , `)
= Prob[h1 ≤ `]
ρ(E )
≥p
≥1
j
We show that (for all p):
E →∞
C (`|E ) −−−−√
−→ F
` E
` − `∗ (E )
σ (E )
S. Sabhapandit (LPTMS, Orsay, France)
or
lim C `∗ + σ z |E = F (z )
E →∞
Limit shapes and largest part of Young diagrams
14 / 22
Largest part of Young diagrams: result
minimal difference partitions of E
ρ(E , `) := # such that the largest part h ≤ `
|
{z
} 1
hj
≥p
ρ(E , ` → ∞) = ρ(E ) ≡ # partitions
≥p
≥p
Consider uniform measure: 1/ρ(E )
ρ(E , `)
= Prob[h1 ≤ `]
ρ(E )
C (`|E ) :=
≥p
≥1
j
We show that (for all p):
E →∞
C (`|E ) −−−−√
−→ F
` E
√
where `∗ (E ) =
E
b(p)
` − `∗ (E )
σ (E )
√
ln
E
b(p)
2
b2 (p) = π6 − Li2 (1/y ∗ ) −
or
S. Sabhapandit (LPTMS, Orsay, France)
√
and σ (E ) =
p
(ln y ∗ )2
2
lim C `∗ + σ z |E = F (z )
E →∞
E
b(p)
where y ∗ − y ∗1−p = 1
Limit shapes and largest part of Young diagrams
14 / 22
Largest part of Young diagrams: result
0.4
minimal difference partitions of E
ρ(E , `) := # such that the largest part h ≤ `
1
dF (z )
dz
0.3
ρ(E , ` → ∞) = ρ(E ) ≡ # partitions
0.2
Consider uniform measure: 1/ρ(E )
ρ(E , `)
= Prob[h1 ≤ `]
ρ(E )
C (`|E ) :=
0.1
-2
E →∞
C (`|E ) −−−−√
−→ F
` E
√
where `∗ (E ) =
E
b(p)
` − `∗ (E )
σ (E )
√
ln
E
b(p)
2
b2 (p) = π6 − Li2 (1/y ∗ ) −
0
2
4
6
Z
We show that (for all p):
or
√
and σ (E ) =
p
(ln y ∗ )2
2
lim C `∗ + σ z |E = F (z )
E →∞
E
b(p)
where y ∗ − y ∗1−p = 1
The scaling function has the Gumbel form: F (z ) = exp − exp(−z )
Earlier result existed only for the p = 0 case [Erdös & Lehner (1951)]
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
14 / 22
Largest part of Young diagrams: derivation of the result
β→0
−−→
ln Z (β, `) −
| {z }
1
β
Z
β`
ln yp e− d
where
yp (x ) − x yp1−p (x ) = 1
0
Derivation
X
e−β E ρ(E , `)
E
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
15 / 22
Largest part of Young diagrams: derivation of the result
1
β→0
−−→
ln Z (β, `) −
β
| {z }
Z
β`
ln yp e− d
where
yp (x ) − x yp1−p (x ) = 1
0
Derivation
X
e−β E ρ(E , `)
E
Therefore formally inverting (in the limit β → 0):
ρ(E , `) =
1
2π i
Z
γ +i ∞
exp SE ,` (β ) dβ where SE ,` (β ) = β E + ln Z (β, `)
γ−i ∞
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
15 / 22
Largest part of Young diagrams: derivation of the result
1
β→0
−−→
ln Z (β, `) −
β
| {z }
Z
β`
ln yp e− d
where
yp (x ) − x yp1−p (x ) = 1
0
Derivation
X
e−β E ρ(E , `)
E
Therefore formally inverting (in the limit β → 0):
ρ(E , `) =
1
2π i
Z
γ +i ∞
exp SE ,` (β ) dβ where SE ,` (β ) = β E + ln Z (β, `)
γ−i ∞
To leading order (using saddle point appximation):
C
ρ(E , `) ≈ exp SE ,` (β ∗ )
and
S. Sabhapandit (LPTMS, Orsay, France)
∂ SE ,` (β ) =0
∂β β =β ∗
Limit shapes and largest part of Young diagrams
15 / 22
Largest part of Young diagrams: derivation of the result
Z
1
β→0
−−→
ln Z (β, `) −
β
| {z }
β`
ln yp e− d
where
yp (x ) − x yp1−p (x ) = 1
0
Derivation
X
e−β E ρ(E , `)
E
Therefore formally inverting (in the limit β → 0):
ρ(E , `) =
Z
1
γ +i ∞
exp SE ,` (β ) dβ where SE ,` (β ) = β E + ln Z (β, `)
2π i
γ−i ∞
To leading order (using saddle point appximation):
C
D
ρ(E , `) ≈ exp SE ,` (β ∗ )
h
√ i2
β∗ E =
Z
β∗ `
and
∂ SE ,` (β ) =0
∂β β =β ∗
∗
ln yp e− d − β ∗ ` ln yp e−β `
0
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
15 / 22
Largest part of Young diagrams: derivation of the result
β
| {z }
β`
Z
1
β→0
−−→
ln Z (β, `) −
ln yp e− d
where
yp (x ) − x yp1−p (x ) = 1
0
Derivation
X
e−β E ρ(E , `)
E
Therefore formally inverting (in the limit β → 0):
ρ(E , `) =
Z
1
γ +i ∞
exp SE ,` (β ) dβ where SE ,` (β ) = β E + ln Z (β, `)
2π i
γ−i ∞
To leading order (using saddle point appximation):
C
D
ρ(E , `) ≈ exp SE ,` (β ∗ )
h
√ i2
β∗ E =
β∗ `
Z
and
∂ SE ,` (β ) =0
∂β β =β ∗
∗
ln yp e− d − β ∗ ` ln yp e−β `
0
E
∗
SE ,` (β ) =
1
β∗
" Z
2
β∗ `
ln yp e
−
∗
d − β ` ln yp e
−β ∗ `
#
0
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
15 / 22
Largest part of Young diagrams: derivation of the result
β
| {z }
β`
Z
1
β→0
−−→
ln Z (β, `) −
ln yp e− d
where
yp (x ) − x yp1−p (x ) = 1
0
Derivation
X
e−β E ρ(E , `)
E
Therefore formally inverting (in the limit β → 0):
ρ(E , `) =
Z
1
γ +i ∞
exp SE ,` (β ) dβ where SE ,` (β ) = β E + ln Z (β, `)
2π i
γ−i ∞
To leading order (using saddle point appximation):
C
D
ρ(E , `) ≈ exp SE ,` (β ∗ )
h
√ i2
β∗ E =
β∗ `
Z
and
∂ SE ,` (β ) =0
∂β β =β ∗
∗
ln yp e− d − β ∗ ` ln yp e−β `
0
E
∗
SE ,` (β ) =
1
β∗
" Z
2
β∗ `
ln yp e
−
∗
d − β ` ln yp e
−β ∗ `
#
0
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
15 / 22
Largest part of Young diagrams: derivation of the result
A
C
B
ρ(E , `) ≈ exp SE ,` (β ∗ )
SE ,` (β ) =
h
" Z
1
∗
β∗
Z
√ i2
β∗ E =
2
yp (x ) − x yp1−p (x ) = 1
β∗ `
ln yp e
−
∗
d − β ` ln yp e
−β ∗ `
#
0
β∗ `
∗
ln yp e− d − β ∗ ` ln yp e−β `
0
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
16 / 22
Largest part of Young diagrams: derivation of the result
A
C
B
ρ(E , `) ≈ exp SE ,` (β ∗ )
SE ,` (β ) =
h
" Z
1
∗
β∗
Z
√ i2
β∗ E =
2
yp (x ) − x yp1−p (x ) = 1
β∗ `
ln yp e
−
∗
d − β ` ln yp e
−β ∗ `
#
0
β∗ `
∗
ln yp e− d − β ∗ ` ln yp e−β `
√
β ∗ ∼ 1/ E
0
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
16 / 22
Largest part of Young diagrams: derivation of the result
A
C
B
ρ(E , `) ≈ exp SE ,` (β ∗ )
SE ,` (β ) =
h
β
√ i2
∗
E
" Z
1
∗
β∗
Z
=
2
yp (x ) − x yp1−p (x ) = 1
β∗ `
ln yp e
−
∗
d − β ` ln yp e
−β ∗ `
0
β∗ `
∗
ln yp e− d − β ∗ ` ln yp e−β `
#
√
=
E gp
`
√
E
√
β ∗ ∼ 1/ E
0
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
16 / 22
Largest part of Young diagrams: derivation of the result
A
C
B
ρ(E , `) ≈ exp SE ,` (β ∗ )
SE ,` (β ) =
h
β
√ i2
∗
E
" Z
1
∗
β∗
Z
=
2
yp (x ) − x yp1−p (x ) = 1
β∗ `
ln yp e
−
∗
d − β ` ln yp e
−β ∗ `
0
β∗ `
∗
ln yp e− d − β ∗ ` ln yp e−β `
#
√
=
E gp
`
√
E
√
β ∗ ∼ 1/ E
0
`
√
E
= x and β ∗ ` = Hp (x )
√
Hp (x )
⇒ β∗ E =
S. Sabhapandit (LPTMS, Orsay, France)
x
Limit shapes and largest part of Young diagrams
16 / 22
Largest part of Young diagrams: derivation of the result
A
C
B
ρ(E , `) ≈ exp SE ,` (β ∗ )
SE ,` (β ) =
h
β
√ i2
∗
E
" Z
1
∗
β∗ `
2
β∗
Z
yp (x ) − x yp1−p (x ) = 1
ln yp e
−
∗
d − β ` ln yp e
−β ∗ `
#
0
β∗ `
=
∗
ln yp e− d − β ∗ ` ln yp e−β `
√
=
E gp
`
√
E
√
β ∗ ∼ 1/ E
0
`
√
E
B
Hp (x )
2
x
C
A
√
Hp (x )
⇒ β∗ E =
= x and β ∗ ` = Hp (x )
gp (x ) = 2
Z
Hp (x )
x
ln yp e− d − Hp (x ) ln yp e−Hp (x )
=
0
Hp (x )
x
ρ(E , `) ≈ exp
+ x ln yp e−Hp (x )
√
E gp
`
√
E
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
16 / 22
Largest part of Young diagrams: derivation of the result
B
Hp (x )
2
x
C
A
gp (x ) = 2
Z
Hp (x )
ln yp e− d − Hp (x ) ln yp e−Hp (x )
=
0
Hp (x )
x
ρ(E , `) ≈ exp
+ x ln yp e−Hp (x )
√
E gp
`
√
yp (x ) − x yp1−p (x ) = 1
E
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
17 / 22
Largest part of Young diagrams: derivation of the result
B
Hp (x )
2
x
C
A
`
gp (x ) = 2
Z
Hp (x )
ln yp e− d − Hp (x ) ln yp e−Hp (x )
=
0
Hp (x )
x
ρ(E , `) ≈ exp
+ x ln yp e−Hp (x )
√
E gp
`
√
yp (x ) − x yp1−p (x ) = 1
E
√
E (large x )
1
gp (x ) ≈ 2b(p) −
exp [−b(p)x ]
b(p)
S. Sabhapandit (LPTMS, Orsay, France)
where b2 (p) =
Z
∞
ln yp e− d
0
Limit shapes and largest part of Young diagrams
17 / 22
Largest part of Young diagrams: derivation of the result
B
Hp (x )
2
x
C
A
`
gp (x ) = 2
Z
Hp (x )
ln yp e− d − Hp (x ) ln yp e−Hp (x )
=
0
Hp (x )
x
ρ(E , `) ≈ exp
+ x ln yp e−Hp (x )
√
E gp
`
√
yp (x ) − x yp1−p (x ) = 1
E
√
E (large x )
1
gp (x ) ≈ 2b(p) −
exp [−b(p)x ]
b(p)
h
where b2 (p) =
Z
∞
ln yp e− d
0
√ i
⇒ ρ(E ) = ρ(E , ` → ∞) ∼ exp 2b(p) E
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
17 / 22
Largest part of Young diagrams: derivation of the result
B
Hp (x )
2
x
C
A
`
gp (x ) = 2
Z
Hp (x )
ln yp e− d − Hp (x ) ln yp e−Hp (x )
=
0
Hp (x )
x
ρ(E , `) ≈ exp
+ x ln yp e−Hp (x )
√
E gp
`
√
yp (x ) − x yp1−p (x ) = 1
E
√
E (large x )
1
gp (x ) ≈ 2b(p) −
exp [−b(p)x ]
b(p)
h
where b2 (p) =
Z
∞
ln yp e− d
0
√ i
⇒ ρ(E ) = ρ(E , ` → ∞) ∼ exp 2b(p) E
√
ρ(E , `)
E
b(p)
` − `∗ (E )
≈ exp −
exp − √ `
= exp − exp − √
ρ(E )
b(p)
E
E /b(p)
√
√
`∗ (E ) =
E /b(p) ln
E /b(p)
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
17 / 22
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Exclusion statistics
Integer partitions
Integer partitions
ideal bosons/fermions
Minimal difference partitions
exclusion statistics
On minimal difference partitions
1
Limit shapes of Young diagrams
1
2
3
4
2
Largest part of Young diagrams
1
2
3
3
Earlier results
Our results
Derivation
Physical interpretation
Our result & earlier result
Derivation
Restricted grand partition function
Summary
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
1 / 22
Restricted grand partition function
ρ(E , N , `, s) := #
=
X
{hj }
minimal difference partitions of E into N parts
such that the largest part ≤ ` and the smallest part ≥ s
δ
E−
N
X
hj
! "N − 1
#
Y
·
θ hj − hj+1 − p · θ ` − h1 · θ hN − s
j=1
S. Sabhapandit (LPTMS, Orsay, France)
i=1
Limit shapes and largest part of Young diagrams
18 / 22
Restricted grand partition function
ρ(E , N , `, s) := #
=
X
minimal difference partitions of E into N parts
such that the largest part ≤ ` and the smallest part ≥ s
δ
E−
{hj }
N
X
hj
! "N − 1
#
Y
·
θ hj − hj+1 − p · θ ` − h1 · θ hN − s
j=1
∞
X
i=1
N=0
z
}|
{
β→0
−−→
ln Z (β, z , `, s) −
1
β
Z
zN
X
e−β E ρ(E , N , `, s)
E
β`
ln yp z e− d
where
yp (x ) − x yp1−p (x ) = 1.
βs
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
18 / 22
Restricted grand partition function
ρ(E , N , `, s) := #
=
X
minimal difference partitions of E into N parts
such that the largest part ≤ ` and the smallest part ≥ s
δ
E−
{hj }
N
X
hj
! "N − 1
#
Y
·
θ hj − hj+1 − p · θ ` − h1 · θ hN − s
j=1
∞
X
i=1
zN
N=0
z
}|
{
β→0
−−→
ln Z (β, z , `, s) −
1
β
Z
X
e−β E ρ(E , N , `, s)
E
β`
ln yp z e− d
where
yp (x ) − x yp1−p (x ) = 1.
βs
Derivation
ρ(E , N , `, s) − ρ(E , N , ` − 1, s) = ρ(E − `, N − 1, ` − p , s)
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
18 / 22
Restricted grand partition function
ρ(E , N , `, s) := #
=
X
minimal difference partitions of E into N parts
such that the largest part ≤ ` and the smallest part ≥ s
δ
E−
{hj }
N
X
hj
! "N − 1
#
Y
·
θ hj − hj+1 − p · θ ` − h1 · θ hN − s
j=1
∞
X
i=1
zN
N=0
z
}|
{
β→0
−−→
ln Z (β, z , `, s) −
1
β
Z
X
e−β E ρ(E , N , `, s)
E
β`
ln yp z e− d
where
yp (x ) − x yp1−p (x ) = 1.
βs
Derivation
ρ(E , N , `, s) − ρ(E , N , ` − 1, s) = ρ(E − `, N − 1, ` − p , s)
Z (β, z , `, s) = Z (β, z , ` − 1, s) + z e−β` Z (β, z , ` − p , s)
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
18 / 22
Restricted grand partition function
ρ(E , N , `, s) := #
=
X
minimal difference partitions of E into N parts
such that the largest part ≤ ` and the smallest part ≥ s
δ
E−
{hj }
N
X
hj
! "N − 1
#
Y
·
θ hj − hj+1 − p · θ ` − h1 · θ hN − s
j=1
∞
X
i=1
zN
N=0
z
}|
{
β→0
−−→
ln Z (β, z , `, s) −
1
β
Z
X
e−β E ρ(E , N , `, s)
E
β`
ln yp z e− d
where
yp (x ) − x yp1−p (x ) = 1.
βs
Derivation
ρ(E , N , `, s) − ρ(E , N , ` − 1, s) = ρ(E − `, N − 1, ` − p , s)
Z (β, z , `, s) = Z (β, z , ` − 1, s) + z e−β` Z (β, z , ` − p , s)
Ansatz:
Z (β, z , `, s) ≈ exp β −1 Φ(β`, β s , z )
as β → 0.
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
18 / 22
Restricted grand partition function
ρ(E , N , `, s) := #
=
X
minimal difference partitions of E into N parts
such that the largest part ≤ ` and the smallest part ≥ s
δ
E−
{hj }
N
X
hj
! "N − 1
#
Y
·
θ hj − hj+1 − p · θ ` − h1 · θ hN − s
j=1
∞
X
i=1
zN
N=0
z
}|
{
β→0
−−→
ln Z (β, z , `, s) −
1
β
Z
X
e−β E ρ(E , N , `, s)
E
β`
ln yp z e− d
where
yp (x ) − x yp1−p (x ) = 1.
βs
Derivation
ρ(E , N , `, s) − ρ(E , N , ` − 1, s) = ρ(E − `, N − 1, ` − p , s)
Z (β, z , `, s) = Z (β, z , ` − 1, s) + z e−β` Z (β, z , ` − p , s)
Ansatz:
Z (β, z , `, s) ≈ exp β −1 Φ(β`, β s , z )
as β → 0.
Substituting in the recursion and Taylor expanding φ up to 1st order
exp(−Φβ` ) + z e−β` exp(−pΦβ` ) = 1, where
S. Sabhapandit (LPTMS, Orsay, France)
Φβ` =
∂
Φ(u , β s , z )
u=β`
∂u
Limit shapes and largest part of Young diagrams
18 / 22
Restricted grand partition function
ρ(E , N , `, s) := #
=
X
minimal difference partitions of E into N parts
such that the largest part ≤ ` and the smallest part ≥ s
δ
E−
{hj }
N
X
hj
! "N − 1
#
Y
·
θ hj − hj+1 − p · θ ` − h1 · θ hN − s
j=1
∞
X
i=1
zN
N=0
z
}|
{
β→0
−−→
ln Z (β, z , `, s) −
1
β
Z
X
e−β E ρ(E , N , `, s)
E
β`
ln yp z e− d
where
yp (x ) − x yp1−p (x ) = 1.
βs
Derivation
ρ(E , N , `, s) − ρ(E , N , `, s + 1) = ρ(E − s , N − 1, `, s + p)
Ansatz:
Z (β, z , `, s) ≈ exp β −1 Φ(β`, β s , z )
as
β → 0.
Substituting in the recursion and Taylor expanding φ up to 1st order
exp(−Φβ` ) + z e−β` exp(−pΦβ` ) = 1, where
S. Sabhapandit (LPTMS, Orsay, France)
Φβ` =
∂
Φ(u , β s , z )
u=β`
∂u
Limit shapes and largest part of Young diagrams
18 / 22
Restricted grand partition function
ρ(E , N , `, s) := #
=
X
minimal difference partitions of E into N parts
such that the largest part ≤ ` and the smallest part ≥ s
δ
E−
{hj }
N
X
hj
! "N − 1
#
Y
·
θ hj − hj+1 − p · θ ` − h1 · θ hN − s
j=1
∞
X
i=1
zN
N=0
z
}|
{
β→0
−−→
ln Z (β, z , `, s) −
1
β
Z
X
e−β E ρ(E , N , `, s)
E
β`
ln yp z e− d
where
yp (x ) − x yp1−p (x ) = 1.
βs
Derivation
ρ(E , N , `, s) − ρ(E , N , `, s + 1) = ρ(E − s , N − 1, `, s + p)
Z (β, z , `, s) = Z (β, z , `, s + 1) + z e−β s Z (β, z , `, s + p)
Ansatz:
Z (β, z , `, s) ≈ exp β −1 Φ(β`, β s , z )
as β → 0.
Substituting in the recursion and Taylor expanding φ up to 1st order
∂
Φ(u , β s , z )
∂u
u=β`
∂
=
Φ(β`, v , z )
v =β s
∂v
exp(−Φβ` ) + z e−β` exp(−pΦβ` ) = 1, where
Φβ` =
exp(Φβ s ) + z e−β s exp(pΦβ s ) = 1,
Φβ s
S. Sabhapandit (LPTMS, Orsay, France)
where
Limit shapes and largest part of Young diagrams
18 / 22
Restricted grand partition function
ρ(E , N , `, s) := #
=
X
minimal difference partitions of E into N parts
such that the largest part ≤ ` and the smallest part ≥ s
δ
E−
{hj }
N
X
hj
! "N − 1
#
Y
·
θ hj − hj+1 − p · θ ` − h1 · θ hN − s
j=1
∞
X
i=1
zN
N=0
z
}|
{
β→0
−−→
ln Z (β, z , `, s) −
1
β
Z
X
e−β E ρ(E , N , `, s)
E
β`
ln yp z e− d
where
yp (x ) − x yp1−p (x ) = 1.
βs
Derivation
Z (β, z , `, s) = Z (β, z , `, s + 1) + z e−β s Z (β, z , `, s + p)
Ansatz:
Z (β, z , `, s) ≈ exp β −1 Φ(β`, β s , z )
as β → 0.
Substituting in the recursion and Taylor expanding φ up to 1st order
∂
Φ(u , β s , z )
∂u
u=β`
∂
−β s
exp(Φβ s ) + z e
exp(pΦβ s ) = 1,
where Φβ s =
Φ(β`, v , z )
v =β s
∂v
−β`
−β
s
Solutions are:
Φβ` = ln yp z e
and Φβ s = − ln yp z e
.
exp(−Φβ` ) + z e−β` exp(−pΦβ` ) = 1, where
S. Sabhapandit (LPTMS, Orsay, France)
Φβ` =
Limit shapes and largest part of Young diagrams
yp (x ) satisfies
ρ(E , N , `, s) − ρ(E , N , `, s + 1) = ρ(E − s , N − 1, `, s + p)
18 / 22
Restricted grand partition function
ρ(E , N , `, s) := #
=
X
minimal difference partitions of E into N parts
such that the largest part ≤ ` and the smallest part ≥ s
δ
E−
{hj }
N
X
hj
! "N − 1
#
Y
·
θ hj − hj+1 − p · θ ` − h1 · θ hN − s
j=1
∞
X
i=1
zN
N=0
z
}|
{
β→0
−−→
ln Z (β, z , `, s) −
1
β
Z
X
e−β E ρ(E , N , `, s)
E
β`
ln yp z e− d
where
yp (x ) − x yp1−p (x ) = 1.
βs
Derivation
Φ(u , v , z ) =
Z
v
u
Z (β, z , `, s) = Z (β, z , `, s + 1) + z e−β s Z (β, z , `, s + p)
Ansatz:
Z (β, z , `, s) ≈ exp β −1 Φ(β`, β s , z )
as β → 0.
Substituting in the recursion and Taylor expanding φ up to 1st order
∂
Φ(u , β s , z )
∂u
u=β`
∂
−β s
exp(Φβ s ) + z e
exp(pΦβ s ) = 1,
where Φβ s =
Φ(β`, v , z )
v =β s
∂v
−β`
−β
s
Solutions are:
Φβ` = ln yp z e
and Φβ s = − ln yp z e
.
exp(−Φβ` ) + z e−β` exp(−pΦβ` ) = 1, where
S. Sabhapandit (LPTMS, Orsay, France)
Φβ` =
Limit shapes and largest part of Young diagrams
yp (x ) satisfies
ln yp z e−
d
ρ(E , N , `, s) − ρ(E , N , `, s + 1) = ρ(E − s , N − 1, `, s + p)
18 / 22
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Exclusion statistics
Integer partitions
Integer partitions
ideal bosons/fermions
Minimal difference partitions
exclusion statistics
On minimal difference partitions
1
Limit shapes of Young diagrams
1
2
3
4
2
Largest part of Young diagrams
1
2
3
3
Earlier results
Our results
Derivation
Physical interpretation
Our result & earlier result
Derivation
Restricted grand partition function
Summary
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
1 / 22
Summary
1
hj
Minimal difference partitions
≥p
E =
X
hj
≥p
j
such that hj − hj+1 ≥ p for j = 1, 2, 3 . . .
h
Wh
≥p
≥p
≥1
S. Sabhapandit (LPTMS, Orsay, France)
j
Limit shapes and largest part of Young diagrams
19 / 22
Summary
1
hj
Minimal difference partitions
≥p
E =
X
hj
≥p
j
h
such that hj − hj+1 ≥ p for j = 1, 2, 3 . . .
Wh
≥p
≥p
2
Partitions
≥1
exclusion statistics
β→0
ln Z (β, z , `, s) −
−−→
1
β
Z
j
β`
ln yp z e
−
d
where
yp (x ) − x yp1−p (x ) = 1.
βs
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
19 / 22
Summary
1
hj
Minimal difference partitions
≥p
E =
X
hj
≥p
j
h
such that hj − hj+1 ≥ p for j = 1, 2, 3 . . .
≥p
Wh
≥p
2
Partitions
β→0
ln Z (β, z , `, s) −
−−→
3
≥1
exclusion statistics
1
β
Z
j
β`
ln yp z e
−
d
where
yp (x ) − x yp1−p (x ) = 1.
βs
Limit shapes of Young diagrams (and physical interpretation)
X=
1
b(p)
ln yp e−b(p)Y
h
or
Y =−
√ i
lim Wh / E
E →∞
h
1
b(p)
ln 1 − e−b(p)X − pX
√ i
lim h/ E
E →∞
S. Sabhapandit (LPTMS, Orsay, France)
b2 (p) =
R∞
0
ln yp e− d
Limit shapes and largest part of Young diagrams
19 / 22
Summary
1
hj
Minimal difference partitions
≥p
E =
X
hj
≥p
j
h
such that hj − hj+1 ≥ p for j = 1, 2, 3 . . .
≥p
Wh
≥p
2
Partitions
β→0
ln Z (β, z , `, s) −
−−→
3
1
β
Z
j
β`
ln yp z e
−
d
yp (x ) − x yp1−p (x ) = 1.
where
βs
Limit shapes of Young diagrams (and physical interpretation)
X=
1
b(p)
ln yp e−b(p)Y
h
or
Y =−
√ i
1
b(p)
ln 1 − e−b(p)X − pX
lim h/ E
E →∞
√ i
h
lim Wh / E
4
≥1
exclusion statistics
b2 (p) =
E →∞
R∞
0
ln yp e− d
Asymptotic distribution of the largest part of the Young diagram
√
√
b(p)
E
E
Distribution of z = √
`−
ln
b(p)
b(p)
E
S. Sabhapandit (LPTMS, Orsay, France)
E →∞
Gumbel distribution
F (z ) = exp (− exp(−z ))
Limit shapes and largest part of Young diagrams
19 / 22
References
Haldane F D M
Fractional statistics in arbitrary dimensions: a generalization of the
Pauli principle
Phys. Rev. Lett. 67, 937 (1991).
Wu Y S
Statistical distribution for generalized ideal gas of fractional statistics
particles
Phys. Rev. Lett. 73, 922 (1994).
Polychronakos A P
Probabilities and path-integral realization of exclusion statistics
Phys. Lett. B 365, 202 (1996).
de Veigy A D and Ouvry S
Equation of state of an anyon gas in a strong magnetic field
Phys. Rev. Lett. 72, 600 (1994).
Murthy M V N and Shankar R
Thermodynamics of a one-dimensional ideal gas with fractional
exclusion statistics
Phys. Rev. Lett. 73 3331 (1994).
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
20 / 22
References
(cont.)
Isakov S B
Fractional statistics in one dimension: modeling by means of 1/x 2
interaction and statistical mechanics
Int. J. Mod. Phys. A 9 2563 (1994).
Hardy G H and Ramanujan S
Asymptotic formulæ in combinatory analysis
Proc. London. Math. Soc. 17, 75 (1918).
Abramowitz M and Stegun I A (Eds.)
Partitions into Distinct Parts
§24.2.2 in Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables, 9th printing, pp. 825-826,
(New York: Dover, 1972).
Comtet A, Majumdar S N and Ouvry S
Integer partitions and exclusion statistics
J. Phys. A: Math. Theor. 40, 11255 (2007).
Temperley H N Y
Statistical mechanics and the partition of numbers: the form of
crystal surfaces
Proc. Cambridge Philos. Soc. 48, 683 (1952).
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
21 / 22
References
(cont.)
Vershik A M
Statistical mechanics of combinatorial partitions and their limit shapes
Funct. Anal. Appl. 30, 90 (1996).
Freiman G, Vershik A M and Yakubovich Yu V
A local limit theorem for random strict partitions
Theory Probab. Appl 44, 453 (2000).
Vershik A M and Yakubovich Yu V
The limit shape and fluctuations of random partitions of naturals with
fixed number of summands
Moscow Math. J. 1, 457 (2001).
Romik D
Identities arising from limit shapes of constrained random partitions
Preprint (2003).
Erdös P and Lehner J
The distribution of the number of summands in the partitions of a
positive integer
Duke Math. J. 8 335 (1951).
S. Sabhapandit (LPTMS, Orsay, France)
Limit shapes and largest part of Young diagrams
22 / 22