Integer partitions and exclusion statistics
Limit shapes and the largest part of Young diagrams
Sanjib Sabhapandit
Raman Research Institute, Bangalore
ICTS-NESP2010 @ IITK, 8 February 2010
Collaborators
Alain Comtet
Satya N. Majumdar
Stéphane Ouvry
References
1. J. Stat. Mech. (2007) P10001
2. J. Math. Phys. Anal. Geom. 4, 1 (2007)
Convex Polygons and their shapes
Taken from [Rajesh & Dhar (2005)]
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
1 / 19
Interface between ordered phases
Change in height
Total particle current
Taken from [Krapivsky, Redner, & Tailleur (2004)]
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
2 / 19
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Integer partitions
Integer partitions
Exclusion statistics
Exclusion statistics
minimal difference partitions
On minimal difference partitions
1
Grand partition function
2
Limit shapes of Young diagrams
1
2
3
3
3
ideal bosons/fermions
Earlier results
Our results
Physical interpretation
Largest part of Young diagrams
Summary
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
3 / 19
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Integer partitions
Integer partitions
Exclusion statistics
Exclusion statistics
minimal difference partitions
On minimal difference partitions
1
Grand partition function
2
Limit shapes of Young diagrams
1
2
3
3
3
ideal bosons/fermions
Earlier results
Our results
Physical interpretation
Largest part of Young diagrams
Summary
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
3 / 19
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Integer partitions
Integer partitions
Exclusion statistics
Exclusion statistics
minimal difference partitions
On minimal difference partitions
1
Grand partition function
2
Limit shapes of Young diagrams
1
2
3
3
3
ideal bosons/fermions
Earlier results
Our results
Physical interpretation
Largest part of Young diagrams
Summary
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
3 / 19
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Integer partitions
Integer partitions
Exclusion statistics
Exclusion statistics
minimal difference partitions
On minimal difference partitions
1
Grand partition function
2
Limit shapes of Young diagrams
1
2
3
3
3
ideal bosons/fermions
Earlier results
Our results
Physical interpretation
Largest part of Young diagrams
Summary
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
3 / 19
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Integer partitions
Integer partitions
Exclusion statistics
Exclusion statistics
minimal difference partitions
On minimal difference partitions
1
Grand partition function
2
Limit shapes of Young diagrams
1
2
3
3
3
ideal bosons/fermions
Earlier results
Our results
Physical interpretation
Largest part of Young diagrams
Summary
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
3 / 19
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Integer partitions
Integer partitions
Exclusion statistics
Exclusion statistics
minimal difference partitions
On minimal difference partitions
1
Grand partition function
2
Limit shapes of Young diagrams
1
2
3
3
3
ideal bosons/fermions
Earlier results
Our results
Physical interpretation
Largest part of Young diagrams
Summary
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
3 / 19
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
4 / 19
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 }.
4=4
ρ(4) = 5
ρ(5) = 7
=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 (RRI, Bangalore)
ρ(10) = 42
ρ(100) = 190569292
..
.
ρ(E ) ≈
1
4E
√ eπ
√
2E /3
3
[Hardy & Ramanujan (1918)]
Limit shapes and largest part of Young diagrams
4 / 19
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 }.
4=4
ρ(4) = 5
ρ(5) = 7
=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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
4 / 19
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 }.
4=4
ρ(4) = 5
ρ(5) = 7
=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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
4 / 19
Integer partitions
Bose/Fermi statistics
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
5 / 19
Integer partitions
18
ideal bosons/fermions
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
6 / 19
Integer partitions
18
ideal bosons/fermions
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
5
3
hj ≥ hj+1
ni = 0, 1, . . . , ∞ (bosons).
3
j
91 = 18 + 16 + 13 + 13 + 9
+6+5+5+3+3
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
6 / 19
Integer partitions
18
ideal bosons/fermions
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
6 / 19
Integer partitions
18
ideal bosons/fermions
ni := #{columns whose heights = i }.
16
E =
13 13
X
hj =
Number of parts N =
9
with i = i .
ni 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 (RRI, Bangalore)
=
X
{ni }
δ
E−
∞
X
!
ni i
i=1
Limit shapes and largest part of Young diagrams
6 / 19
Integer partitions
18
ideal bosons/fermions
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 (RRI, Bangalore)
{ni }
i=1
!
δ
N−
∞
X
!
ni
i=1
Limit shapes and largest part of Young diagrams
6 / 19
Integer partitions
ideal bosons/fermions
18
ni := #{columns whose heights = i }.
16
E =
13 13
X
hj =
Number of parts N =
9
with i = i .
ni 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
+6+5+5+3+3
{ni }
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
6 / 19
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
micro-canonical partition function
| {z }
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
7 / 19
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)
micro-canonical partition function
| {z }
E
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
7 / 19
Exclusion statistics
Minimal difference partitions
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
8 / 19
Minimal difference p partitions
hj
≥p
≥p
E =
X
hj
such that
hj − hj+1 ≥ p
j
p = 0, 1, 2, . . .
≥p
≥p
≥s
ρ(E , N , `, s) := #
=
X
{hj }
j
minimal difference partitions of E into N parts
such that the largest part ≤ ` and the smallest part ≥ s
δ
E−
N
X
! "N −1
#
Y
hj ·
θ hj − hj+1 − p · θ ` − h1 · θ hN − s
j=1
S. Sabhapandit (RRI, Bangalore)
i=1
Limit shapes and largest part of Young diagrams
9 / 19
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Integer partitions
Integer partitions
Exclusion statistics
Exclusion statistics
minimal difference partitions
On minimal difference partitions
1
Grand partition function
2
Limit shapes of Young diagrams
1
2
3
3
3
ideal bosons/fermions
Earlier results
Our results
Physical interpretation
Largest part of Young diagrams
Summary
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
3 / 19
Grand partition function
hj
minimal difference partitions
 of E into N parts such that 
ρ(E , N , `, s) := # 

the largest part ≤ `, and
the smallest part ≥ s


= ρ(E , N , ` − 1, s) + ρ(E − `, N − 1, ` − p , s)
= ρ(E , N , `, s + 1) + ρ(E − s , N − 1, `, s + p)
≥p
≥p
≥p
≥p
≥s
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
j
4 / 19
Grand partition function
hj
minimal difference partitions
 of E into N parts such that 
ρ(E , N , `, s) := # 

the largest part ≤ `, and
the smallest part ≥ s


= ρ(E , N , ` − 1, s) + ρ(E − `, N − 1, ` − p , s)
= ρ(E , N , `, s + 1) + ρ(E − s , N − 1, `, s + p)
≥p
≥p
≥p
≥p
≥s
Z (β, z , `, s) : =
∞
X
N=0
zN
X
j
e−β E ρ(E , N , `, s)
E
= Z (β, z , ` − 1, s) + z e−β` Z (β, z , ` − p , s)
= Z (β, z , `, s + 1) + z e−β s Z (β, z , `, s + p)
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
4 / 19
Grand partition function
hj
minimal difference partitions
 of E into N parts such that 
ρ(E , N , `, s) := # 

the largest part ≤ `, and
the smallest part ≥ s


≥p
≥p
≥p
= ρ(E , N , ` − 1, s) + ρ(E − `, N − 1, ` − p , s)
= ρ(E , N , `, s + 1) + ρ(E − s , N − 1, `, s + p)
≥p
≥s
Z (β, z , `, s) : =
∞
X
zN
N=0
X
j
e−β E ρ(E , N , `, s)
E
= Z (β, z , ` − 1, s) + z e−β` Z (β, z , ` − p , 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 )
β→0
ln Z (β, z , `, s) −
−−→
1
β
Z
β`
ln yp z e− d
as
where
β → 0.
yp (x ) − x yp1−p (x ) = 1.
βs
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
4 / 19
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Integer partitions
Integer partitions
Exclusion statistics
Exclusion statistics
minimal difference partitions
On minimal difference partitions
1
Grand partition function
2
Limit shapes of Young diagrams
1
2
3
3
3
ideal bosons/fermions
Earlier results
Our results
Physical interpretation
Largest part of Young diagrams
Summary
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
3 / 19
Limit shapes
hj
≥p
≥p
h
≥p
Wh
≥p
≥1
j
Let
X = lim
E →∞
Y = lim
E →∞
Wh
√
,
E
h
√
E
.
The limit shape is given by
the XY curve.
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
10 / 19
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
X = lim
E →∞
Y = lim
E →∞
Wh
√
,
E
h
√
E
.
The limit shape is given by
the XY curve.
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
10 / 19
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
10 / 19
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
X = lim
E →∞
Y = lim
E →∞
Wh
√
√
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 (RRI, Bangalore)
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 / 19
Derivation of the limit shapes
Wh = number of columns
whose heights ≥ h.
hj
≥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.
S. Sabhapandit (RRI, Bangalore)
≥p
h
Wh
≥p
≥p
≥1
j
Limit shapes and largest part of Young diagrams
11 / 19
Derivation of the limit shapes
Wh = number of columns
whose heights ≥ h.
hj
≥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
≥p
h
Wh
≥p
≥p
≥1
j
∂
h Wh i = z
ln Zh (β, z )
z =1
∂z
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
11 / 19
Derivation of the limit shapes
Wh = number of columns
whose heights ≥ h.
hj
≥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
≥p
h
Wh
≥p
≥p
≥1
j
∂
h Wh i = z
ln Zh (β, z )
z =1
∂z
z
}|
{
∂
E =−
ln Z (β, 1)
∂β
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
11 / 19
Derivation of the limit shapes
Wh = number of columns
whose heights ≥ h.
hj
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
≥p
h
≥p
≥p
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
Wh
≥1
j
∂
h Wh 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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
11 / 19
Derivation of the limit shapes
Wh = number of columns
whose heights ≥ h.
hj
≥p
Zh (β, z ) := the restricted grand partition
function which counts the columns
whose heights ≥ h.
≥p
h
A
≥p
≥p
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
∂
h Wh i = z
ln Zh (β, z )
z =1
∂z
Wh
≥1
β hWh i = ln yp e−β h
j
z
}|
{
∂
E =−
ln Z (β, 1)
∂β
B
hWh2 i − hWh i2 =
z
∂ ∂
z
ln Zh (β, z )
z =1
∂z ∂z
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
11 / 19
Derivation of the limit shapes
Wh = number of columns
whose heights ≥ h.
hj
≥p
Zh (β, z ) := the restricted grand partition
≥p
function which counts the columns
whose heights ≥ h.
h
}|
{
∂
E =−
ln Z (β, 1)
∂β
B
≥1
∂
h Wh i = z
ln Zh (β, z )
z =1
∂z
z
hWh2 i − hWh i2 =
≥p
≥p
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
Wh
β hWh i = ln yp e−β h
j
b(p)
β= √
z
E
∂ ∂
z
ln Zh (β, z )
z =1
∂z ∂z
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
11 / 19
Derivation of the limit shapes
Wh = number of columns
whose heights ≥ h.
hj
≥p
Zh (β, z ) := the restricted grand partition
≥p
function which counts the columns
whose heights ≥ h.
h
}|
{
∂
E =−
ln Z (β, 1)
∂β
B
≥1
∂
h Wh i = z
ln Zh (β, z )
z =1
∂z
z
≥p
≥p
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
Wh
β hWh i = ln yp e−β h
j
b(p)
β= √
E
"
#
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
11 / 19
Derivation of the limit shapes
Wh = number of columns
whose heights ≥ h.
hj
≥p
Zh (β, z ) := the restricted grand partition
≥p
function which counts the columns
whose heights ≥ h.
h
≥1
∂
h Wh i = z
ln Zh (β, z )
z =1
∂z
}|
{
∂
E =−
ln Z (β, 1)
∂β
β hWh i = ln yp e−β h
z
B
≥p
≥p
Z (β, z ) = Z1 (β, z ) is the full grand
partition function which counts all the
columns.
A
Wh
j
b(p)
β= √
E
"
#
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 (RRI, Bangalore)
Wh
√
E
h E →∞
vs √ −−−→ Limit shape.
E
Limit shapes and largest part of Young diagrams
11 / 19
Our (general) formulæ for limit shapes
A
X =
1
b(p)
ln yp e
−b(p)Y
hj
X = lim
E →∞
≥p
or
B
Wh
√
Y =−
Y = lim
1
b(p)
h
in which
yp (w ) − w yp1−p (w ) = 1
Z
ln yp e
−
d
0
=
π2
6
− Li2 (1/y ∗ ) −
where y ∗ = yp (1)
and Li2 (z ) =
∞
X
k=1
p
z
k2
E
2
j
π
(ln y ∗ )2
b(0) = √
6
π
i.e., y ∗ − y ∗1−p = 1
k
√
≥p
≥1
∞
b (p) =
Wh
E
h
≥p
and
2
E →∞
≥p
ln 1 − e−b(p)X − pX
b(1) = √
12
π
b(2) = √
15
is the dilogarithm function. b(3) = 0.752617 . . .
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
12 / 19
Our (general) formulæ for limit shapes
A
X =
1
b(p)
ln yp e
−b(p)Y
b(p)Y
3
X = lim
E →∞
or
B
Y =−
Y = lim
1
ln 1 − e−b(p)X − pX
b(p)
E →∞
2
Wh
√
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
− Li2 (1/y ∗ ) −
where y ∗ = yp (1)
and Li2 (z ) =
∞
X
k=1
p
2
z
k2
2
3
π
(ln y ∗ )2
b(0) = √
6
π
i.e., y ∗ − y ∗1−p = 1
k
1
b(1) = √
12
π
b(2) = √
15
is the dilogarithm function. b(3) = 0.752617 . . .
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
12 / 19
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
j
13 / 19
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
j
13 / 19
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 (RRI, Bangalore)
hj
h
Limit shapes and largest part of Young diagrams
13 / 19
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
13 / 19
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
13 / 19
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 (RRI, Bangalore)
hj
Limit shapes and largest part of Young diagrams
13 / 19
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 (RRI, Bangalore)
hj
Limit shapes and largest part of Young diagrams
13 / 19
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 (RRI, Bangalore)
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 / 19
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Integer partitions
Integer partitions
Exclusion statistics
Exclusion statistics
minimal difference partitions
On minimal difference partitions
1
Grand partition function
2
Limit shapes of Young diagrams
1
2
3
3
3
ideal bosons/fermions
Earlier results
Our results
Physical interpretation
Largest part of Young diagrams
Summary
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
3 / 19
Largest part of Young diagrams
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
≥s
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
j
14 / 19
Largest part of Young diagrams
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 (RRI, Bangalore)
≥p
≥p
Consider uniform measure: 1/ρ(E )
C (`|E ) :=
hj
≥p
Limit shapes and largest part of Young diagrams
≥s
j
14 / 19
Largest part of Young diagrams
minimal difference partitions of E
ρ(E , `) := # such that the largest part h ≤ `
|
{z
} 1
ρ(E , ` → ∞) = ρ(E ) ≡ # partitions
≥p
≥p
Consider uniform measure: 1/ρ(E )
C (`|E ) :=
hj
≥p
ρ(E , `)
= Prob[h1 ≤ `]
ρ(E )
≥p
≥s
j
We show that (for all p):
E →∞
C (`|E ) −−−−√
−→ F
` E
` − `∗ (E )
σ (E )
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
14 / 19
Largest part of Young diagrams
minimal difference partitions of E
ρ(E , `) := # such that the largest part h ≤ `
|
{z
} 1
hj
≥p
ρ(E , ` → ∞) = ρ(E ) ≡ # partitions
≥p
Consider uniform measure: 1/ρ(E )
C (`|E ) :=
≥p
ρ(E , `)
= Prob[h1 ≤ `]
ρ(E )
≥p
≥s
j
We show that (for all p):
E →∞
C (`|E ) −−−−√
−→ F
` E
` − `∗ (E )
σ (E )
S. Sabhapandit (RRI, Bangalore)
or
lim C `∗ + σ z |E = F (z )
E →∞
Limit shapes and largest part of Young diagrams
14 / 19
Largest part of Young diagrams
minimal difference partitions of E
ρ(E , `) := # such that the largest part h ≤ `
|
{z
} 1
hj
≥p
ρ(E , ` → ∞) = ρ(E ) ≡ # partitions
≥p
Consider uniform measure: 1/ρ(E )
≥p
ρ(E , `)
= Prob[h1 ≤ `]
ρ(E )
C (`|E ) :=
≥p
≥s
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 (RRI, Bangalore)
√
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 / 19
Largest part of Young diagrams
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
0
-2
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 ∗ ) −
2
4
6
Z
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
14 / 19
Outline
1
Integer partitions and exclusion statistics
1
2
3
4
2
Integer partitions
Integer partitions
Exclusion statistics
Exclusion statistics
minimal difference partitions
On minimal difference partitions
1
Grand partition function
2
Limit shapes of Young diagrams
1
2
3
3
3
ideal bosons/fermions
Earlier results
Our results
Physical interpretation
Largest part of Young diagrams
Summary
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
3 / 19
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 (RRI, Bangalore)
j
Limit shapes and largest part of Young diagrams
15 / 19
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
β`
ln yp z e
−
d
where
j
yp (x ) − x yp1−p (x ) = 1.
βs
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
15 / 19
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
β`
ln yp z e
−
d
where
j
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
√ i
lim Wh / E
h
E →∞
or
Y =−
h
1
b(p)
ln 1 − e−b(p)X − pX
√ i
lim h/ E
E →∞
S. Sabhapandit (RRI, Bangalore)
b2 (p) =
R∞
0
ln yp e− d
Limit shapes and largest part of Young diagrams
15 / 19
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
β`
ln yp z e
−
yp (x ) − x yp1−p (x ) = 1.
where
d
j
βs
Limit shapes of Young diagrams (and physical interpretation)
X=
1
b(p)
ln yp e−b(p)Y
or
√ i
lim Wh / E
h
Y =−
1
b(p)
ln 1 − e−b(p)X − pX
√ i
h
lim h/ E
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 (RRI, Bangalore)
E →∞
Gumbel distribution
F (z ) = exp (− exp(−z ))
Limit shapes and largest part of Young diagrams
15 / 19
References
Rajesh and Dhar
Convex lattice polygons of fixed area with perimeter-dependent
weights
Phys. Rev. E 71, 016130 (2005).
P L Krapivsky, S Redner, and J Tailleur.
Dynamics of an unbounded interface between ordered phases
Phys. Rev. E 69, 026125 (2004).
http://link.aps.org/abstract/PRE/v69/e026125
Haldane F D M
Fractional statistics in arbitrary dimensions: a generalization of the
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Wu Y S
Statistical distribution for generalized ideal gas of fractional statistics
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Phys. Rev. Lett. 73, 922 (1994).
Polychronakos A P
Probabilities and path-integral realization of exclusion statistics
Phys. Lett. B 365, 202 (1996).
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
16 / 19
References
(cont.)
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).
Isakov S B
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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).
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
17 / 19
References
(cont.)
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).
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).
S. Sabhapandit (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
18 / 19
References
(cont.)
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 (RRI, Bangalore)
Limit shapes and largest part of Young diagrams
19 / 19