Outline Introduction Burke property Tropical RSK
The exactly solvable log-gamma polymer
Timo Seppäläinen
Department of Mathematics
University of Wisconsin-Madison
2011
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
1
Introduction
2
Burke property
3
Tropical RSK
Collaborators: Nicos Georgiou (Utah), Ivan Corwin (Microsoft/MIT), Neil
O’Connell (Warwick), Nikos Zygouras (Warwick)
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Context. Among directed polymer models there are currently two 1+1
dimensional exactly solvable cases:
polymer in a Brownian environment with continuous-time random
walk paths, discovered by O’Connell-Yor (2001), subsequently worked
on by O’Connell, O’C–Moriarty, and Valkó–S.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Context. Among directed polymer models there are currently two 1+1
dimensional exactly solvable cases:
polymer in a Brownian environment with continuous-time random
walk paths, discovered by O’Connell-Yor (2001), subsequently worked
on by O’Connell, O’C–Moriarty, and Valkó–S.
lattice polymer whose weights are i.i.d. log-gamma distributed
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Context. Among directed polymer models there are currently two 1+1
dimensional exactly solvable cases:
polymer in a Brownian environment with continuous-time random
walk paths, discovered by O’Connell-Yor (2001), subsequently worked
on by O’Connell, O’C–Moriarty, and Valkó–S.
lattice polymer whose weights are i.i.d. log-gamma distributed
This talk is about the log-gamma polymer.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
1+1 dimensional lattice polymer with fixed endpoints
n
6
•••
••
••
•••
•
•
1 ••
1
m
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
1+1 dimensional lattice polymer with fixed endpoints
• Πm,n = set of up-right lattice paths x = (xk )
n
6
•••
••
••
•••
•
•
1 ••
1
from (1, 1) to (m, n)
m
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
1+1 dimensional lattice polymer with fixed endpoints
• Πm,n = set of up-right lattice paths x = (xk )
n
6
•••
••
••
•••
•
•
1 ••
1
from (1, 1) to (m, n)
• environment of i.i.d. weights under P:
{ω(x) : x ∈ N2 }
m
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
1+1 dimensional lattice polymer with fixed endpoints
• Πm,n = set of up-right lattice paths x = (xk )
n
6
•••
••
••
•••
•
•
1 ••
1
m
Log-gamma polymer Warwick September 2011
from (1, 1) to (m, n)
• environment of i.i.d. weights under P:
{ω(x) : x ∈ N2 }
• quenched probability measure on Πm,n :
n P
o
1
exp β k ω(xk )
Qm,n (x ) =
Zm,n
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Outline Introduction Burke property Tropical RSK
1+1 dimensional lattice polymer with fixed endpoints
• Πm,n = set of up-right lattice paths x = (xk )
n
6
•••
••
••
•••
•
•
1 ••
1
m
from (1, 1) to (m, n)
• environment of i.i.d. weights under P:
{ω(x) : x ∈ N2 }
• quenched probability measure on Πm,n :
n P
o
1
exp β k ω(xk )
Qm,n (x ) =
Zm,n
• inverse temperature β > 0
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
1+1 dimensional lattice polymer with fixed endpoints
• Πm,n = set of up-right lattice paths x = (xk )
n
from (1, 1) to (m, n)
6
•••
••
••
•••
•
•
1 ••
1
• environment of i.i.d. weights under P:
{ω(x) : x ∈ N2 }
• quenched probability measure on Πm,n :
n P
o
1
exp β k ω(xk )
Qm,n (x ) =
Zm,n
m
• inverse temperature β > 0
• partition function Zm,n =
X
n P
o
exp β k ω(xk )
x ∈Πm,n
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Outline Introduction Burke property Tropical RSK
Key quantities again:
Quenched measure Qm,n (x ) =
Partition function Zm,n =
Log-gamma polymer Warwick September 2011
1
Zm,n
n X
o
exp β
ω(xk )
k
X
n X
o
exp β
ω(xk )
x ∈Πm,n
k
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Outline Introduction Burke property Tropical RSK
Key quantities again:
Quenched measure Qm,n (x ) =
Partition function Zm,n =
1
Zm,n
n X
o
exp β
ω(xk )
k
X
n X
o
exp β
ω(xk )
x ∈Πm,n
k
Questions:
Behavior of walk x under Qm,n on large scales.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Key quantities again:
Quenched measure Qm,n (x ) =
Partition function Zm,n =
1
Zm,n
n X
o
exp β
ω(xk )
k
X
n X
o
exp β
ω(xk )
x ∈Πm,n
k
Questions:
Behavior of walk x under Qm,n on large scales.
Behavior of log Zm,n (now also random as a function of ω).
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Key quantities again:
Quenched measure Qm,n (x ) =
Partition function Zm,n =
1
Zm,n
n X
o
exp β
ω(xk )
k
X
n X
o
exp β
ω(xk )
x ∈Πm,n
k
Questions:
Behavior of walk x under Qm,n on large scales.
Behavior of log Zm,n (now also random as a function of ω).
In this talk focus is on log Z .
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Specialize to log-gamma distribution
Fix β = 1.
Pick a parameter 0 < µ < ∞.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Specialize to log-gamma distribution
Fix β = 1.
Pick a parameter 0 < µ < ∞.
Let Yi, j = e ω(i,j) be i.i.d. Gamma−1 (µ) distributed,
in other words, Yi,−1
j ∼ Gamma(µ).
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Specialize to log-gamma distribution
Fix β = 1.
Pick a parameter 0 < µ < ∞.
Let Yi, j = e ω(i,j) be i.i.d. Gamma−1 (µ) distributed,
in other words, Yi,−1
j ∼ Gamma(µ).
Gamma(µ) density: f (x) = Γ(µ)−1 x µ−1 e −x on R+
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Specialize to log-gamma distribution
Fix β = 1.
Pick a parameter 0 < µ < ∞.
Let Yi, j = e ω(i,j) be i.i.d. Gamma−1 (µ) distributed,
in other words, Yi,−1
j ∼ Gamma(µ).
Gamma(µ) density: f (x) = Γ(µ)−1 x µ−1 e −x on R+
E(log Y ) = −Ψ0 (µ)
(digamma function)
Var(log Y ) = Ψ1 (µ)
(trigamma function)
where Ψn (s) = (d n+1 /ds n+1 ) log Γ(s)
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Specialize to log-gamma distribution
Fix β = 1.
Pick a parameter 0 < µ < ∞.
Let Yi, j = e ω(i,j) be i.i.d. Gamma−1 (µ) distributed,
in other words, Yi,−1
j ∼ Gamma(µ).
Gamma(µ) density: f (x) = Γ(µ)−1 x µ−1 e −x on R+
E(log Y ) = −Ψ0 (µ)
(digamma function)
Var(log Y ) = Ψ1 (µ)
(trigamma function)
where Ψn (s) = (d n+1 /ds n+1 ) log Γ(s)
What is special about this choice?
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Stationary version of the model
Parameters 0 < θ < µ.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Stationary version of the model
Parameters 0 < θ < µ.
Bulk weights Yi, j for i, j ∈ N = {1, 2, 3, . . . }
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Stationary version of the model
Parameters 0 < θ < µ.
Bulk weights Yi, j for i, j ∈ N = {1, 2, 3, . . . }
Boundary weights Ui,0 = Yi,0 and V0, j = Y0, j .
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Stationary version of the model
Parameters 0 < θ < µ.
Bulk weights Yi, j for i, j ∈ N = {1, 2, 3, . . . }
Boundary weights Ui,0 = Yi,0 and V0, j = Y0, j .
.. 6
.
1
0
V0, j
Yi, j
1
Ui,0
0
1
2
Log-gamma polymer Warwick September 2011
...
-
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Outline Introduction Burke property Tropical RSK
Stationary version of the model
Parameters 0 < θ < µ.
Bulk weights Yi, j for i, j ∈ N = {1, 2, 3, . . . }
Boundary weights Ui,0 = Yi,0 and V0, j = Y0, j .
Yi,−1
j ∼ Gamma(µ)
.. 6
.
1
0
V0, j
Yi, j
1
Ui,0
0
1
2
Log-gamma polymer Warwick September 2011
−1
Ui,0
∼ Gamma(θ)
...
-
V0,−1j ∼ Gamma(µ − θ)
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Outline Introduction Burke property Tropical RSK
In 2-parameter model, compute Zm,n ∀ (m, n) ∈ Z2+ and define
Um,n =
Zm,n
Zm−1,n
Vm,n =
Log-gamma polymer Warwick September 2011
Zm,n
Zm,n−1
Xm,n =
Z
Zm,n −1
m,n
+
Zm+1,n
Zm,n+1
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Outline Introduction Burke property Tropical RSK
In 2-parameter model, compute Zm,n ∀ (m, n) ∈ Z2+ and define
Um,n =
Zm,n
Zm−1,n
Vm,n =
For an undirected edge f :
Log-gamma polymer Warwick September 2011
Zm,n
Zm,n−1
(
Ux
Tf =
Vx
Xm,n =
Z
Zm,n −1
m,n
+
Zm+1,n
Zm,n+1
f = {x − e1 , x}
f = {x − e2 , x}
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Outline Introduction Burke property Tropical RSK
In 2-parameter model, compute Zm,n ∀ (m, n) ∈ Z2+ and define
Um,n =
Zm,n
Zm−1,n
Vm,n =
For an undirected edge f :
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
• • ••
• • ••
Log-gamma polymer Warwick September 2011
Zm,n
Zm,n−1
Xm,n =
(
Ux
Tf =
Vx
Z
Zm,n −1
m,n
+
Zm+1,n
Zm,n+1
f = {x − e1 , x}
f = {x − e2 , x}
down-right path (zk ) with
edges fk = {zk−1 , zk }, k ∈ Z
•
interior points I of path (zk )
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Outline Introduction Burke property Tropical RSK
In 2-parameter model, compute Zm,n ∀ (m, n) ∈ Z2+ and define
Um,n =
Zm,n
Zm−1,n
Vm,n =
For an undirected edge f :
••
Xz P
••
q
P
••
••
••
••
• Vx
•
Ux
•
•
• • ••
• • ••
Log-gamma polymer Warwick September 2011
Zm,n
Zm,n−1
Xm,n =
(
Ux
Tf =
Vx
Z
Zm,n −1
m,n
+
Zm+1,n
Zm,n+1
f = {x − e1 , x}
f = {x − e2 , x}
down-right path (zk ) with
edges fk = {zk−1 , zk }, k ∈ Z
•
interior points I of path (zk )
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Outline Introduction Burke property Tropical RSK
Burke property
Theorem
Variables {Tfk , Xz : k ∈ Z, z ∈ I } are independent with marginals
U −1 ∼ Gamma(θ), V −1 ∼ Gamma(µ − θ), and X −1 ∼ Gamma(µ).
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Burke property
Theorem
Variables {Tfk , Xz : k ∈ Z, z ∈ I } are independent with marginals
U −1 ∼ Gamma(θ), V −1 ∼ Gamma(µ − θ), and X −1 ∼ Gamma(µ).
“Burke property” because the analogous property for last-passage
percolation with exponential weights is a generalization of Burke’s
Theorem for M/M/1 queues.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Consequences of Burke property: free energy density
Partition functions
θ
Zm,n for i.i.d. Γ−1 (µ) model, Zm,n
for stationary (θ, µ)-model.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Consequences of Burke property: free energy density
Partition functions
θ
Zm,n for i.i.d. Γ−1 (µ) model, Zm,n
for stationary (θ, µ)-model.
Limits: p(s, t) = limn→∞
Log-gamma polymer Warwick September 2011
θ
log Zns,nt
log Zns,nt
and p θ (s, t) = lim
n→∞
n
n
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Outline Introduction Burke property Tropical RSK
Consequences of Burke property: free energy density
Partition functions
θ
Zm,n for i.i.d. Γ−1 (µ) model, Zm,n
for stationary (θ, µ)-model.
Limits: p(s, t) = limn→∞
bntc
θ
log Zns,nt
log Zns,nt
and p θ (s, t) = lim
n→∞
n
n
θ
log Zns,nt
=
nt
X
j=1
log V0, j +
ns
X
log Ui,nt
i=1
bnsc
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Consequences of Burke property: free energy density
Partition functions
θ
Zm,n for i.i.d. Γ−1 (µ) model, Zm,n
for stationary (θ, µ)-model.
Limits: p(s, t) = limn→∞
bntc
θ
log Zns,nt
log Zns,nt
and p θ (s, t) = lim
n→∞
n
n
θ
log Zns,nt
=
nt
X
j=1
bnsc
Log-gamma polymer Warwick September 2011
log V0, j +
ns
X
log Ui,nt
i=1
hence p θ (s, t) = −tΨ0 (µ − θ) − sΨ0 (θ)
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Outline Introduction Burke property Tropical RSK
Now to compute p(s, t):
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Now to compute p(s, t):
bntc
6
•••
••
••
` • •••
•
1 •
0 •
0 1 2
...
bnsc
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Now to compute p(s, t):
bntc
6
•••
••
••
` • •••
•
1 •
0 •
0 1 2
...
bnsc
Log-gamma polymer Warwick September 2011
θ
Zns,nt
=
ns Y
k
X
k=1
+
Ui,0
Z(k,1),(ns,nt)
i=1
nt Y
`
X
`=1
V0, j
Z(1,`),(ns,nt)
j=1
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Outline Introduction Burke property Tropical RSK
Now to compute p(s, t):
bntc
6
•••
••
••
` • •••
•
1 •
0 •
0 1 2
...
θ
Zns,nt
=
ns Y
k
X
k=1
+
bnsc
p θ (s, t) = sup {−aΨ0 (θ)+p(s −a, t)}
0≤a≤s
Log-gamma polymer Warwick September 2011
Ui,0
Z(k,1),(ns,nt)
i=1
nt Y
`
X
`=1
_
V0, j
Z(1,`),(ns,nt)
j=1
sup {−bΨ0 (µ−θ)+p(s, t−b)}
0≤b≤t
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Outline Introduction Burke property Tropical RSK
Now to compute p(s, t):
bntc
6
•••
••
••
` • •••
•
1 •
0 •
0 1 2
...
θ
Zns,nt
=
ns Y
k
X
k=1
+
bnsc
p θ (s, t) = sup {−aΨ0 (θ)+p(s −a, t)}
0≤a≤s
Ui,0
Z(k,1),(ns,nt)
i=1
nt Y
`
X
`=1
_
V0, j
Z(1,`),(ns,nt)
j=1
sup {−bΨ0 (µ−θ)+p(s, t−b)}
0≤b≤t
Vary θ. Specializes to a convex duality that can be inverted to give
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Now to compute p(s, t):
bntc
6
•••
••
••
` • •••
•
1 •
0 •
0 1 2
...
θ
Zns,nt
=
ns Y
k
X
k=1
+
bnsc
0≤a≤s
_
Ui,0
Z(k,1),(ns,nt)
i=1
nt Y
`
X
`=1
p θ (s, t) = sup {−aΨ0 (θ)+p(s −a, t)}
V0, j
Z(1,`),(ns,nt)
j=1
sup {−bΨ0 (µ−θ)+p(s, t−b)}
0≤b≤t
Vary θ. Specializes to a convex duality that can be inverted to give
p(s, t) = inf {−sΨ0 (θ) − tΨ0 (µ − θ)}
0<θ<µ
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Fluctuation bounds
Burke property also serves as basis for deriving fluctuation exponents.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Fluctuation bounds
Burke property also serves as basis for deriving fluctuation exponents.
If endpoint (m, n) → ∞ in characteristic direction
| m − NΨ1 (µ − θ) | ≤ CN 2/3
and | n − NΨ1 (θ) | ≤ CN 2/3
then fluctuations have conjectured order of magnitude:
θ
N 1/3 for log Zm,n
Log-gamma polymer Warwick September 2011
and
N 2/3
for the path.
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Outline Introduction Burke property Tropical RSK
Fluctuation bounds
Burke property also serves as basis for deriving fluctuation exponents.
If endpoint (m, n) → ∞ in characteristic direction
| m − NΨ1 (µ − θ) | ≤ CN 2/3
and | n − NΨ1 (θ) | ≤ CN 2/3
then fluctuations have conjectured order of magnitude:
θ
N 1/3 for log Zm,n
and
N 2/3
for the path.
θ
Further away from the characteristic log Zm,n
satisfies CLT.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Fluctuation bounds
Burke property also serves as basis for deriving fluctuation exponents.
If endpoint (m, n) → ∞ in characteristic direction
| m − NΨ1 (µ − θ) | ≤ CN 2/3
and | n − NΨ1 (θ) | ≤ CN 2/3
then fluctuations have conjectured order of magnitude:
θ
N 1/3 for log Zm,n
and
N 2/3
for the path.
θ
Further away from the characteristic log Zm,n
satisfies CLT.
Upper bounds hold for i.i.d. model without boundaries.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Explicit large deviations for log Z
L.m.g.f. of log Y , Y ∼ Γ−1 (µ):
Mµ (ξ) = log E e
ξ log Y
Log-gamma polymer Warwick September 2011
=
(
log Γ(µ − ξ) − log Γ(µ)
∞
ξ ∈ (−∞, µ)
ξ ∈ [µ, ∞).
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Outline Introduction Burke property Tropical RSK
Explicit large deviations for log Z
L.m.g.f. of log Y , Y ∼ Γ−1 (µ):
Mµ (ξ) = log E e
ξ log Y
=
(
log Γ(µ − ξ) − log Γ(µ)
∞
ξ ∈ (−∞, µ)
ξ ∈ [µ, ∞).
For i.i.d. Γ−1 (µ) model, let
Λs,t (ξ) = lim n−1 log E e ξ log Zns,nt ,
n→∞
Log-gamma polymer Warwick September 2011
ξ∈R
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Outline Introduction Burke property Tropical RSK
Explicit large deviations for log Z
L.m.g.f. of log Y , Y ∼ Γ−1 (µ):
Mµ (ξ) = log E e
ξ log Y
=
(
log Γ(µ − ξ) − log Γ(µ)
∞
ξ ∈ (−∞, µ)
ξ ∈ [µ, ∞).
For i.i.d. Γ−1 (µ) model, let
Λs,t (ξ) = lim n−1 log E e ξ log Zns,nt ,
n→∞
ξ∈R
Theorem.


p(s, t)ξ



Λs,t (ξ) =
inf tMθ (ξ) − sMµ−θ (−ξ)
θ∈(ξ,µ)


∞
Log-gamma polymer Warwick September 2011
ξ<0
0≤ξ<µ
ξ ≥ µ.
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Outline Introduction Burke property Tropical RSK
Λs,t linear on R− because for r < p(s, t)
lim n−1 log P{log Zns,nt ≤ nr } = −∞.
n→∞
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Λs,t linear on R− because for r < p(s, t)
lim n−1 log P{log Zns,nt ≤ nr } = −∞.
n→∞
Right tail LDP: for r ≥ p(s, t)
Js,t (r ) ≡ − lim n−1 log P{log Zns,nt ≥ nr } = Λ∗s,t (r )
n→∞
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Λs,t linear on R− because for r < p(s, t)
lim n−1 log P{log Zns,nt ≤ nr } = −∞.
n→∞
Right tail LDP: for r ≥ p(s, t)
Js,t (r ) ≡ − lim n−1 log P{log Zns,nt ≥ nr } = Λ∗s,t (r )
n→∞
Proof of formula for Λs,t goes by first finding Js,t and then convex
conjugation.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Starting point for proof of large deviations
bntc
`
1
0
6
•••
••
••
• •••
•
•
•
0 1
θ
Zns,nt
=
nt Y
`
X
`=1
+
V0, j
Z(1,`),(ns,nt)
j=1
ns Y
k
X
k=1
Ui,0
Z(k,1),(ns,nt)
i=1
bnsc
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Starting point for proof of large deviations
bntc
`
1
0
6
•••
••
••
• •••
•
•
•
-
=
nt Y
`
X
`=1
+
V0, j
Z(1,`),(ns,nt)
j=1
ns Y
k
X
k=1
Ui,0
Z(k,1),(ns,nt)
i=1
bnsc
0 1
Divide by
θ
Zns,nt
Qnt
ns
Y
i=1
j=1
V0, j :
Ui,nt =
nt Y
nt
X
`=1
V0,−1j Z(1,`),(ns,nt)
j=`+1
+
ns Y
nt
X
k=1
Log-gamma polymer Warwick September 2011
j=1
V0,−1j
Y
k
Ui,0 Z(k,1),(ns,nt)
i=1
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Outline Introduction Burke property Tropical RSK
Starting point for proof of large deviations
bntc
`
1
0
6
•••
••
••
• •••
•
•
•
-
=
nt Y
`
X
`=1
+
V0, j
Z(1,`),(ns,nt)
j=1
ns Y
k
X
k=1
Ui,0
Z(k,1),(ns,nt)
i=1
bnsc
0 1
Divide by
θ
Zns,nt
Qnt
ns
Y
i=1
j=1
V0, j :
Ui,nt =
nt Y
nt
X
`=1
V0,−1j Z(1,`),(ns,nt)
j=`+1
+
ns Y
nt
X
k=1
j=1
V0,−1j
Y
k
Ui,0 Z(k,1),(ns,nt)
i=1
Now we know LDP for log(l.h.s) and can extract log Z from the r.h.s.
Log-gamma polymer Warwick September 2011
15/36
Outline Introduction Burke property Tropical RSK
Combinatorial approach to the log-gamma polymer
Log-gamma polymer Warwick September 2011
16/36
Outline Introduction Burke property Tropical RSK
Combinatorial approach to the log-gamma polymer
N
• • • k
• •
• •
• • •
•
•
1 • •
n
1
Log-gamma polymer Warwick September 2011
16/36
Outline Introduction Burke property Tropical RSK
Combinatorial approach to the log-gamma polymer
N
Fix N, let 1 ≤ k ≤ N and n ≥ 1 vary.
• • • k
• •
• •
• • •
•
•
1 • •
n
1
Log-gamma polymer Warwick September 2011
16/36
Outline Introduction Burke property Tropical RSK
Combinatorial approach to the log-gamma polymer
N
• • • k
• •
• •
• • •
•
•
1 • •
n
1
Log-gamma polymer Warwick September 2011
Fix N, let 1 ≤ k ≤ N and n ≥ 1 vary.
Π1n,k = { admissible paths (1, 1) → (n, k) }
16/36
Outline Introduction Burke property Tropical RSK
Combinatorial approach to the log-gamma polymer
N
• • • k
• •
• •
• • •
•
•
1 • •
n
1
Log-gamma polymer Warwick September 2011
Fix N, let 1 ≤ k ≤ N and n ≥ 1 vary.
Π1n,k = { admissible paths (1, 1) → (n, k) }
X
zk,1 (n) =
wt(π) where
π∈Π1n,k
weight wt(π) =
Q
(i, j)∈π
Yi, j
16/36
Outline Introduction Burke property Tropical RSK
Combinatorial approach to the log-gamma polymer
N
• • • k
• •
• •
• • •
•
•
1 • •
n
1
Fix N, let 1 ≤ k ≤ N and n ≥ 1 vary.
Π1n,k = { admissible paths (1, 1) → (n, k) }
X
zk,1 (n) =
wt(π) where
π∈Π1n,k
weight wt(π) =
Q
(i, j)∈π
Yi, j
Π`n,k = { `-tuples π = (π1 , . . . , π` ) of disjoint
paths πj : (1, j) → (n, k − j + 1) }
Log-gamma polymer Warwick September 2011
16/36
Outline Introduction Burke property Tropical RSK
Combinatorial approach to the log-gamma polymer
Fix N, let 1 ≤ k ≤ N and n ≥ 1 vary.
N
• • • k
• •
• •
• • •
•
•
1 • •
n
1
π∈Π1n,k
weight wt(π) =
Q
(i, j)∈π
Yi, j
Π`n,k = { `-tuples π = (π1 , . . . , π` ) of disjoint
N
3
2
1
Π1n,k = { admissible paths (1, 1) → (n, k) }
X
zk,1 (n) =
wt(π) where
•
•
•
•
•
1
••• k
• • • • • • k−1
• • • • • • k−2
•
•
•
•
••••
•
••••••• -
paths πj : (1, j) → (n, k − j + 1) }
n
Log-gamma polymer Warwick September 2011
16/36
Outline Introduction Burke property Tropical RSK
Combinatorial approach to the log-gamma polymer
Fix N, let 1 ≤ k ≤ N and n ≥ 1 vary.
N
• • • k
• •
• •
• • •
•
•
1 • •
n
1
π∈Π1n,k
weight wt(π) =
Q
(i, j)∈π
Yi, j
Π`n,k = { `-tuples π = (π1 , . . . , π` ) of disjoint
N
3
2
1
Π1n,k = { admissible paths (1, 1) → (n, k) }
X
zk,1 (n) =
wt(π) where
•
•
•
•
•
1
••• k
• • • • • • k−1
• • • • • • k−2
•
•
•
•
••••
•
••••••• -
paths πj : (1, j) → (n, k − j + 1) }
Q
weight wt(π) = (i, j)∈π Yi, j
n
Log-gamma polymer Warwick September 2011
16/36
Outline Introduction Burke property Tropical RSK
Combinatorial approach to the log-gamma polymer
Fix N, let 1 ≤ k ≤ N and n ≥ 1 vary.
N
• • • k
• •
• •
• • •
•
•
1 • •
n
1
π∈Π1n,k
weight wt(π) =
Q
(i, j)∈π
Yi, j
Π`n,k = { `-tuples π = (π1 , . . . , π` ) of disjoint
N
3
2
1
Π1n,k = { admissible paths (1, 1) → (n, k) }
X
zk,1 (n) =
wt(π) where
•
•
•
•
•
1
••• k
• • • • • • k−1
• • • • • • k−2
•
•
•
•
••••
•
••••••• n
Log-gamma polymer Warwick September 2011
paths πj : (1, j) → (n, k − j + 1) }
Q
weight wt(π) = (i, j)∈π Yi, j
τk,` (n) =
X
wt(π)
π∈Π`n,k
16/36
Outline Introduction Burke property Tropical RSK
The sum of the weights of the `-tuples of non-intersecting paths
X
τk,` (n) =
wt(π)
for 1 ≤ k ≤ N, 1 ≤ ` ≤ n ∧ k.
π∈Π`n,k
Log-gamma polymer Warwick September 2011
17/36
Outline Introduction Burke property Tropical RSK
The sum of the weights of the `-tuples of non-intersecting paths
X
τk,` (n) =
wt(π)
for 1 ≤ k ≤ N, 1 ≤ ` ≤ n ∧ k.
π∈Π`n,k
Define array z(n) = {zk,` (n) : 1 ≤ k ≤ N, 1 ≤ ` ≤ k ∧ n} by
X
zk,1 (n) · · · zk,` (n) = τk` (n) =
wt(π).
π∈Π`n,k
Log-gamma polymer Warwick September 2011
17/36
Outline Introduction Burke property Tropical RSK
The sum of the weights of the `-tuples of non-intersecting paths
X
τk,` (n) =
wt(π)
for 1 ≤ k ≤ N, 1 ≤ ` ≤ n ∧ k.
π∈Π`n,k
Define array z(n) = {zk,` (n) : 1 ≤ k ≤ N, 1 ≤ ` ≤ k ∧ n} by
X
zk,1 (n) · · · zk,` (n) = τk` (n) =
wt(π).
π∈Π`n,k
N = 4 array
z11 (n)
z22 (n)
z33 (n)
z44 (n)
Log-gamma polymer Warwick September 2011
z21 (n)
z32 (n)
z43 (n)
polymer
z31 (n)
z42 (n)
z41 (n)
17/36
Outline Introduction Burke property Tropical RSK
The sum of the weights of the `-tuples of non-intersecting paths
X
τk,` (n) =
wt(π)
for 1 ≤ k ≤ N, 1 ≤ ` ≤ n ∧ k.
π∈Π`n,k
Define array z(n) = {zk,` (n) : 1 ≤ k ≤ N, 1 ≤ ` ≤ k ∧ n} by
X
zk,1 (n) · · · zk,` (n) = τk` (n) =
wt(π).
π∈Π`n,k
N = 4 array
z11 (n)
z22 (n)
z33 (n)
z44 (n)
Log-gamma polymer Warwick September 2011
z21 (n)
z32 (n)
z43 (n)
polymer
z31 (n)
z42 (n)
z41 (n)
17/36
Outline Introduction Burke property Tropical RSK
The mapping
weight matrix (Yi, j ) 7→ array z(n)
is Kirillov’s tropical RSK correspondence (2001).
Log-gamma polymer Warwick September 2011
RSK
18/36
Outline Introduction Burke property Tropical RSK
The mapping
weight matrix (Yi, j ) 7→ array z(n)
is Kirillov’s tropical RSK correspondence (2001).
RSK
Next develop the Markovian evolution in terms of tropical or geometric
row insertion.
Log-gamma polymer Warwick September 2011
18/36
Outline Introduction Burke property Tropical RSK
The mapping
weight matrix (Yi, j ) 7→ array z(n)
is Kirillov’s tropical RSK correspondence (2001).
RSK
Next develop the Markovian evolution in terms of tropical or geometric
row insertion.
Look at case ` = 1:
• •
•
1
•
1
n−1 n
k
k−1
-
Log-gamma polymer Warwick September 2011
18/36
Outline Introduction Burke property Tropical RSK
The mapping
weight matrix (Yi, j ) 7→ array z(n)
is Kirillov’s tropical RSK correspondence (2001).
RSK
Next develop the Markovian evolution in terms of tropical or geometric
row insertion.
Look at case ` = 1:
• •
•
1
•
1
n−1 n
k
k−1
Add a new column n in weight matrix.
-
Log-gamma polymer Warwick September 2011
18/36
Outline Introduction Burke property Tropical RSK
The mapping
weight matrix (Yi, j ) 7→ array z(n)
is Kirillov’s tropical RSK correspondence (2001).
RSK
Next develop the Markovian evolution in terms of tropical or geometric
row insertion.
Look at case ` = 1:
• •
•
1
•
1
n−1 n
k
k−1
Add a new column n in weight matrix.
z1,1 (n) = Yn,1 z1,1 (n − 1)
-
Log-gamma polymer Warwick September 2011
18/36
Outline Introduction Burke property Tropical RSK
The mapping
weight matrix (Yi, j ) 7→ array z(n)
is Kirillov’s tropical RSK correspondence (2001).
RSK
Next develop the Markovian evolution in terms of tropical or geometric
row insertion.
Look at case ` = 1:
• •
•
k
k−1
Add a new column n in weight matrix.
z1,1 (n) = Yn,1 z1,1 (n − 1)
For k = 2, . . . , N
1
•
1
n−1 n
-
Log-gamma polymer Warwick September 2011
zk,1 (n) = Yn,k zk,1 (n − 1) + zk−1,1 (n)
18/36
Outline Introduction Burke property Tropical RSK
The mapping
weight matrix (Yi, j ) 7→ array z(n)
is Kirillov’s tropical RSK correspondence (2001).
RSK
Next develop the Markovian evolution in terms of tropical or geometric
row insertion.
Look at case ` = 1:
• •
•
k
k−1
Add a new column n in weight matrix.
z1,1 (n) = Yn,1 z1,1 (n − 1)
For k = 2, . . . , N
1
•
1
n−1 n
-
zk,1 (n) = Yn,k zk,1 (n − 1) + zk−1,1 (n)
After this, transformed weights are passed on to diagonal
z2 = (z22 . . . , zN2 ) and that diagonal is updated. And so on.
Log-gamma polymer Warwick September 2011
18/36
Outline Introduction Burke property Tropical RSK
Let 1 ≤ ` ≤ N. Geometric row insertion
of the word b = (b` , . . . , bN ) into the word ξ = (ξ` , . . . , ξN )
produces two new words
0
ξ 0 = (ξ`0 , . . . , ξN
)
0
0
and b 0 = (b`+1
, . . . , bN
).
Notation and definition:
b
ξ
−→
↓
ξ0
where
b0


ξ 0 = b` ξ`

 `


 0
0
ξk = bk (ξk−1
+ ξk ) ` + 1 ≤ k ≤ N


0
 0
ξk ξk−1


` + 1 ≤ k ≤ N.
bk = bk
ξk−1 ξk0
Words have strictly positive real entries.
Log-gamma polymer Warwick September 2011
19/36
Outline Introduction Burke property Tropical RSK
Let z be an array with diagonals z1 , . . . , zN , and b ∈ (0, ∞)N a word.
Log-gamma polymer Warwick September 2011
20/36
Outline Introduction Burke property Tropical RSK
Let z be an array with diagonals z1 , . . . , zN , and b ∈ (0, ∞)N a word.
Geometric row insertion of b into z produces a new array z 0 = z ← b
defined by iterating basic row insertion N times.
Log-gamma polymer Warwick September 2011
20/36
Outline Introduction Burke property Tropical RSK
Let z be an array with diagonals z1 , . . . , zN , and b ∈ (0, ∞)N a word.
Geometric row insertion of b into z produces a new array z 0 = z ← b
defined by iterating basic row insertion N times.
Let a1 = b, and then
a1
z1
−→
↓
z10
a2
z2
−→
↓
z20
a3
..
.
aN
zN
Log-gamma polymer Warwick September 2011
−→
↓
zN0
20/36
Outline Introduction Burke property Tropical RSK
Let z be an array with diagonals z1 , . . . , zN , and b ∈ (0, ∞)N a word.
Geometric row insertion of b into z produces a new array z 0 = z ← b
defined by iterating basic row insertion N times.
Let a1 = b, and then
a1
z1
−→
↓
z10
a2
z2
−→
↓
z20
a3
..
.
aN
zN
−→
↓
zN0
The process exhausts the input, aN+1 is empty.
Log-gamma polymer Warwick September 2011
20/36
Outline Introduction Burke property Tropical RSK
Let z be an array with diagonals z1 , . . . , zN , and b ∈ (0, ∞)N a word.
Geometric row insertion of b into z produces a new array z 0 = z ← b
defined by iterating basic row insertion N times.
Let a1 = b, and then
a1
z1
−→
↓
z10
a2
z2
−→
↓
z20
a3
..
.
aN
zN
−→
↓
zN0
The process exhausts the input, aN+1 is empty.
Diagonals z10 , . . . , zN0 make up the new array z 0 .
Log-gamma polymer Warwick September 2011
20/36
Outline Introduction Burke property Tropical RSK
a1 (1)
z1 (0)
−→
↓
a1 (2)
z1 (1)
a2 (1)
−→
↓
a1 (3)
z1 (2)
a2 (2)
−→
↓
z1 (3)
···
···
a2 (3)
z2 (0)
−→
↓
z2 (1)
−→
↓
z2 (2)
−→
↓
z2 (3)
..
.
a3 (1)
..
.
..
.
a3 (2)
..
.
..
.
a3 (3)
..
.
..
.
aN (1)
zN (0)
−→
↓
aN (2)
zN (1)
−→
↓
aN (3)
zN (2)
−→
↓
zN (3) · · ·
Evolution of the array z(n) over time n = 0, 1, 2, . . .
Log-gamma polymer Warwick September 2011
21/36
Outline Introduction Burke property Tropical RSK
a1 (1)
z1 (0)
−→
↓
a1 (2)
z1 (1)
a2 (1)
−→
↓
a1 (3)
z1 (2)
a2 (2)
−→
↓
z1 (3)
···
···
a2 (3)
z2 (0)
−→
↓
z2 (1)
−→
↓
z2 (2)
−→
↓
z2 (3)
..
.
a3 (1)
..
.
..
.
a3 (2)
..
.
..
.
a3 (3)
..
.
..
.
aN (1)
zN (0)
−→
↓
aN (2)
zN (1)
−→
↓
aN (3)
zN (2)
−→
↓
zN (3) · · ·
Evolution of the array z(n) over time n = 0, 1, 2, . . .
Initial state z(0) is on the left edge in terms of diagonals z1 (0), . . . , zN (0).
Log-gamma polymer Warwick September 2011
21/36
Outline Introduction Burke property Tropical RSK
a1 (1)
z1 (0)
−→
↓
a1 (2)
z1 (1)
a2 (1)
−→
↓
a1 (3)
z1 (2)
a2 (2)
−→
↓
z1 (3)
···
···
a2 (3)
z2 (0)
−→
↓
z2 (1)
−→
↓
z2 (2)
−→
↓
z2 (3)
..
.
a3 (1)
..
.
..
.
a3 (2)
..
.
..
.
a3 (3)
..
.
..
.
aN (1)
zN (0)
−→
↓
aN (2)
zN (1)
−→
↓
aN (3)
zN (2)
−→
↓
zN (3) · · ·
Evolution of the array z(n) over time n = 0, 1, 2, . . .
Initial state z(0) is on the left edge in terms of diagonals z1 (0), . . . , zN (0).
Time n input from weight matrix: a1 (n) = Y [n] = (Yn,1 , . . . , Yn,N ).
At each time, geometric row insertion is iterated N times to update each
diagonal.
Log-gamma polymer Warwick September 2011
21/36
Outline Introduction Burke property Tropical RSK
The previous process is for evolving the full array. We also need the
variant for starting with an empty array z(0) = ∅.
Log-gamma polymer Warwick September 2011
22/36
Outline Introduction Burke property Tropical RSK
The previous process is for evolving the full array. We also need the
variant for starting with an empty array z(0) = ∅.
a1 (1)
(N)
e1
a1 (2)
−→
↓
z1 (1)
−→
↓
a1 (3)
z1 (2)
a2 (2)
(N−1)
e1
−→
↓
−→
↓
z1 (3) · · ·
a2 (3)
z2 (2)
−→
↓
z2 (3) · · ·
a3 (3)
(N−2)
e1
Resulting array:
−→
↓
z3 (3) · · ·
z(n) = ∅ ← Y [1] ← Y [2] ← · · · ← Y [n] .
Log-gamma polymer Warwick September 2011
22/36
Outline Introduction Burke property Tropical RSK
Theorem. The array z(n) defined by Kirillov’s path construction is equal
to z(n) = ∅ ← Y [1] ← Y [2] ← · · · ← Y [n] .
(Noumi and Yamada (2004), proof by a matrix technique.)
Log-gamma polymer Warwick September 2011
23/36
Outline Introduction Burke property Tropical RSK
Theorem. The array z(n) defined by Kirillov’s path construction is equal
to z(n) = ∅ ← Y [1] ← Y [2] ← · · · ← Y [n] .
(Noumi and Yamada (2004), proof by a matrix technique.)
Now let us make the input random.
Log-gamma polymer Warwick September 2011
23/36
Outline Introduction Burke property Tropical RSK
Theorem. The array z(n) defined by Kirillov’s path construction is equal
to z(n) = ∅ ← Y [1] ← Y [2] ← · · · ← Y [n] .
(Noumi and Yamada (2004), proof by a matrix technique.)
Now let us make the input random.
Assumption. Weights {Yn, j } are independent with marginals Yn, j ∼
Γ−1 (θ̂n + θj ) where {θ̂n , θj } are fixed real parameters such that each
θ̂n + θj > 0.
Log-gamma polymer Warwick September 2011
23/36
Outline Introduction Burke property Tropical RSK
Theorem. The array z(n) defined by Kirillov’s path construction is equal
to z(n) = ∅ ← Y [1] ← Y [2] ← · · · ← Y [n] .
(Noumi and Yamada (2004), proof by a matrix technique.)
Now let us make the input random.
Assumption. Weights {Yn, j } are independent with marginals Yn, j ∼
Γ−1 (θ̂n + θj ) where {θ̂n , θj } are fixed real parameters such that each
θ̂n + θj > 0.
Can we say anything about partition function zN,1 (n) ?
Log-gamma polymer Warwick September 2011
23/36
Outline Introduction Burke property Tropical RSK
Theorem. The array z(n) defined by Kirillov’s path construction is equal
to z(n) = ∅ ← Y [1] ← Y [2] ← · · · ← Y [n] .
(Noumi and Yamada (2004), proof by a matrix technique.)
Now let us make the input random.
Assumption. Weights {Yn, j } are independent with marginals Yn, j ∼
Γ−1 (θ̂n + θj ) where {θ̂n , θj } are fixed real parameters such that each
θ̂n + θj > 0.
Can we say anything about partition function zN,1 (n) ?
Markov kernel Πn for time n transition z(n − 1) → z(n) is complicated.
Log-gamma polymer Warwick September 2011
23/36
Outline Introduction Burke property Tropical RSK
Theorem. The array z(n) defined by Kirillov’s path construction is equal
to z(n) = ∅ ← Y [1] ← Y [2] ← · · · ← Y [n] .
(Noumi and Yamada (2004), proof by a matrix technique.)
Now let us make the input random.
Assumption. Weights {Yn, j } are independent with marginals Yn, j ∼
Γ−1 (θ̂n + θj ) where {θ̂n , θj } are fixed real parameters such that each
θ̂n + θj > 0.
Can we say anything about partition function zN,1 (n) ?
Markov kernel Πn for time n transition z(n − 1) → z(n) is complicated.
Bottom row y (n) = (zN,1 (n), zN,2 (n), . . . , zN,N (n)) of the array turns out
to be a more tractable Markov chain.
Log-gamma polymer Warwick September 2011
23/36
Outline Introduction Burke property Tropical RSK
Theorem. The array z(n) defined by Kirillov’s path construction is equal
to z(n) = ∅ ← Y [1] ← Y [2] ← · · · ← Y [n] .
(Noumi and Yamada (2004), proof by a matrix technique.)
Now let us make the input random.
Assumption. Weights {Yn, j } are independent with marginals Yn, j ∼
Γ−1 (θ̂n + θj ) where {θ̂n , θj } are fixed real parameters such that each
θ̂n + θj > 0.
Can we say anything about partition function zN,1 (n) ?
Markov kernel Πn for time n transition z(n − 1) → z(n) is complicated.
Bottom row y (n) = (zN,1 (n), zN,2 (n), . . . , zN,N (n)) of the array turns out
to be a more tractable Markov chain.
Theory of Markov functions shows this.
Log-gamma polymer Warwick September 2011
23/36
Outline Introduction Burke property Tropical RSK
Markov functions idea
Log-gamma polymer Warwick September 2011
24/36
Outline Introduction Burke property Tropical RSK
Markov functions idea
∃ Markov kernel Π for z(n) on space T .
T
Π
?
T
Log-gamma polymer Warwick September 2011
24/36
Outline Introduction Burke property Tropical RSK
Markov functions idea
∃ Markov kernel Π for z(n) on space T .
∃ map φ : T → Y .
T
Π
?
T
Log-gamma polymer Warwick September 2011
φ Y
φ Y
24/36
Outline Introduction Burke property Tropical RSK
Markov functions idea
∃ Markov kernel Π for z(n) on space T .
∃ map φ : T → Y .
When is y (n) = φ(z(n)) Markov with kernel P̄ ?
Log-gamma polymer Warwick September 2011
T
φ Y
Π
P̄ ?
?
T
?
φ Y
24/36
Outline Introduction Burke property Tropical RSK
Markov functions idea
∃ Markov kernel Π for z(n) on space T .
T
∃ map φ : T → Y .
φ Y
Π
When is y (n) = φ(z(n)) Markov with kernel P̄ ?
P̄ ?
?
T
?
φ Y
Sufficient condition. Suppose ∃ (positive but not necessary stochastic)
kernels P : Y → Y and K : Y → T such that
K (y , φ−1 (y )) = 1
Log-gamma polymer Warwick September 2011
and
K ◦Π=P ◦K
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Outline Introduction Burke property Tropical RSK
Markov functions idea
∃ Markov kernel Π for z(n) on space T .
φ Y
T
∃ map φ : T → Y .
Π
When is y (n) = φ(z(n)) Markov with kernel P̄ ?
P̄ ?
?
T
?
φ Y
Sufficient condition. Suppose ∃ (positive but not necessary stochastic)
kernels P : Y → Y and K : Y → T such that
K (y , φ−1 (y )) = 1
and
K ◦Π=P ◦K
Set w (y ) = K (y , T ). Intertwining gives Pw = w . Define stochastic
kernels
K̄ (y , dz) =
1
K (y , dz)
w (y )
Log-gamma polymer Warwick September 2011
and P̄(y , d ỹ ) =
w (ỹ )
P(y , d ỹ )
w (y )
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Outline Introduction Burke property Tropical RSK
Markov functions idea, continued
φ
T - Y
K̄
Then K̄ ◦ Π = P̄ ◦ K̄
P̄
Π
?
T Log-gamma polymer Warwick September 2011
φ Y
K̄
?
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Outline Introduction Burke property Tropical RSK
Markov functions idea, continued
φ
T - Y
K̄
Then K̄ ◦ Π = P̄ ◦ K̄
P̄
Π
?
T φ Y
K̄
?
If z(n) starts with distribution K̄ (y , dz), then y (n) is Markov in its own
filtration with transition P̄ and initial state y (0) = y .
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Outline Introduction Burke property Tropical RSK
Markov functions idea, continued
φ
T - Y
K̄
Then K̄ ◦ Π = P̄ ◦ K̄
P̄
Π
?
T φ Y
K̄
?
If z(n) starts with distribution K̄ (y , dz), then y (n) is Markov in its own
filtration with transition P̄ and initial state y (0) = y .
Furthermore
E f (z(n)) y (0), . . . , y (n − 1), y (n) = y = K̄ f (y )
(Rogers and Pitman, 1981)
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Outline Introduction Burke property Tropical RSK
Application of Markov functions
Spaces: TN = space of arrays of size N,
YN = (0, ∞)N = space of positive N-vectors.
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Outline Introduction Burke property Tropical RSK
Application of Markov functions
Spaces: TN = space of arrays of size N,
YN = (0, ∞)N = space of positive N-vectors.
Define a (substochastic) kernel Pn on YN by
Pn (y , d ỹ ) =
N−1
Y
i=1
ỹi+1
exp −
yi
Log-gamma polymer Warwick September 2011
Y
N j=1
−1
Γ(γn, j )
γn, j
yj
yj d ỹj
exp −
ỹj
ỹj
ỹj
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Outline Introduction Burke property Tropical RSK
Application of Markov functions
Spaces: TN = space of arrays of size N,
YN = (0, ∞)N = space of positive N-vectors.
Define a (substochastic) kernel Pn on YN by
Pn (y , d ỹ ) =
N−1
Y
i=1
ỹi+1
exp −
yi
Y
N −1
Γ(γn, j )
j=1
γn, j
yj
yj d ỹj
exp −
ỹj
ỹj
ỹj
and intertwining kernel K : YN → TN by
K (y , dz) =
Y
1≤`≤k<N
zk,`
θk+1 −θ`
zk+1,`
N
zk,`
zk+1,`+1 dzk,` Y
−
δy` (dzN,` )
× exp −
zk+1,`
zk,`
zk,`
`=1
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Outline Introduction Burke property Tropical RSK
Application of Markov functions
Then K ◦ Πn = Pn ◦ K .
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Outline Introduction Burke property Tropical RSK
Application of Markov functions
Then K ◦ Πn = Pn ◦ K . Bottom row y (n) is a MC with kernel
P̄n (y , d ỹ ) =
w (ỹ )
Pn (y , d ỹ )
w (y )
where w (y ) = K (y , TN ).
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Outline Introduction Burke property Tropical RSK
Application of Markov functions
Then K ◦ Πn = Pn ◦ K . Bottom row y (n) is a MC with kernel
P̄n (y , d ỹ ) =
w (ỹ )
Pn (y , d ỹ )
w (y )
where w (y ) = K (y , TN ).
Conditional distribution of array, given evolution of bottom row:
E f (z(n)) y (0), . . . , y (n − 1), y (n) = y = K̄ f (y )
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Outline Introduction Burke property Tropical RSK
Application of Markov functions
Then K ◦ Πn = Pn ◦ K . Bottom row y (n) is a MC with kernel
P̄n (y , d ỹ ) =
w (ỹ )
Pn (y , d ỹ )
w (y )
where w (y ) = K (y , TN ).
Conditional distribution of array, given evolution of bottom row:
E f (z(n)) y (0), . . . , y (n − 1), y (n) = y = K̄ f (y )
Now a closer look at the eigenfunctions we have found. All the previous
makes sense also for complex parameters. This is beneficial because then
we can use known special functions to diagonalize the transition kernel.
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Outline Introduction Burke property Tropical RSK
Whittaker functions
GL(N, R)-Whittaker function for y ∈ YN , with λ ∈ CN
Ψλ (y ) =
N
Y
i=1
yi−λi
Z
Kλ (y , dz)
TN
where Kλ is the previous intertwining kernel with θ replaced by λ.
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Outline Introduction Burke property Tropical RSK
Whittaker functions
GL(N, R)-Whittaker function for y ∈ YN , with λ ∈ CN
Ψλ (y ) =
N
Y
yi−λi
i=1
Z
Kλ (y , dz)
TN
where Kλ is the previous intertwining kernel with θ replaced by λ.
Intertwining works also with complex parameters and gives
Z
(0,∞)N
Ψθ+λ (ỹ )
P̄n (y , d ỹ ) =
Ψθ (ỹ )
Log-gamma polymer Warwick September 2011
Y
N
i=1
Γ(γn,i + λi ) Ψθ+λ (y )
Γ(γn,i )
Ψθ (y )
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Outline Introduction Burke property Tropical RSK
Our goal is an expression for the distribution of zN,1 (n), the polymer
partition function. Getting there involves two steps.
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Outline Introduction Burke property Tropical RSK
Our goal is an expression for the distribution of zN,1 (n), the polymer
partition function. Getting there involves two steps.
Step 1. Row insertion reveals that if we start with bottom row
y M = (e M(i−(k+1)/2) )1≤i≤N ,
let initial array have distribution K̄ (y N , dz), and let M → ∞, the
distribution of the array z(n) converges to the array started from the
empty array z(0) = ∅.
So this limit recovers the distribution of the partition function from the
path construction.
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Outline Introduction Burke property Tropical RSK
Step 2. Invert the eigenfunction relation to write an expression for the
distribution of bottom row y (n) when started from y ∈ (0, ∞)N . Involves
analytical properties of Whittaker functions (analogous to Fourier
analysis). Take y = y M and let M → ∞.
The result is a formula for the Laplace transform of the partition function:
Z
PN
Y
Γ(λi − θj )
E(e −s zN,1 (n) ) =
s i=1 (θi −λi )
ιRN
1≤i, j≤N
n Y
N
Y
Γ(λi + θ̂m )
m=1 i=1
Γ(θi + θ̂m )
×
sN (λ) dλ
where the Sklyanin measure is given by
sN (λ) =
Y
1
Γ(λj − λk )−1
(2πι)N N!
j6=k
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Outline Introduction Burke property Tropical RSK
A future goal: asymptotics for distribution of log zN,1 (n).
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Outline Introduction Burke property Tropical RSK
Proof of Burke property
Induction on I by flipping a growth corner:
U’
V
•
Y
X
•
U 0 = Y (1 + U/V )
V’
X = U −1 + V −1
V 0 = Y (1 + V /U)
−1
U
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Outline Introduction Burke property Tropical RSK
Proof of Burke property
Induction on I by flipping a growth corner:
U’
V
•
Y
X
•
U 0 = Y (1 + U/V )
V’
X = U −1 + V −1
V 0 = Y (1 + V /U)
−1
U
Lemma. Given that (U, V , Y ) are independent positive r.v.’s,
d
(U 0 , V 0 , X ) = (U, V , Y ) iff (U, V , Y ) have the gamma distr’s.
Proof. “if” part by computation, “only if” part from a characterization
of gamma due to Lukacs (1955).
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Outline Introduction Burke property Tropical RSK
Proof of Burke property
Induction on I by flipping a growth corner:
U’
V
•
Y
X
•
U 0 = Y (1 + U/V )
V’
X = U −1 + V −1
V 0 = Y (1 + V /U)
−1
U
Lemma. Given that (U, V , Y ) are independent positive r.v.’s,
d
(U 0 , V 0 , X ) = (U, V , Y ) iff (U, V , Y ) have the gamma distr’s.
Proof. “if” part by computation, “only if” part from a characterization
of gamma due to Lukacs (1955).
This gives all (zk ) with finite I. General case follows.
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Outline Introduction Burke property Tropical RSK
Combinatorial RSK (Robinson-Schensted-Knuth correspondence)
Classic RSK maps an n × N weight matrix Y = (Yi, j ) with nonnegative
integer entries bijectively to a pair (P, Q) of Young tableaux with common
shape.
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Outline Introduction Burke property Tropical RSK
Combinatorial RSK (Robinson-Schensted-Knuth correspondence)
Classic RSK maps an n × N weight matrix Y = (Yi, j ) with nonnegative
integer entries bijectively to a pair (P, Q) of Young tableaux with common
shape.
1 2
2
2 3
4
Example of (semistandard) Young tableau T .
4 4
5
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Outline Introduction Burke property Tropical RSK
Combinatorial RSK (Robinson-Schensted-Knuth correspondence)
Classic RSK maps an n × N weight matrix Y = (Yi, j ) with nonnegative
integer entries bijectively to a pair (P, Q) of Young tableaux with common
shape.
1 2
2
Example of (semistandard) Young tableau T .
2 3
4
Rows weakly, columns strictly increasing.
4 4
5
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Outline Introduction Burke property Tropical RSK
Combinatorial RSK (Robinson-Schensted-Knuth correspondence)
Classic RSK maps an n × N weight matrix Y = (Yi, j ) with nonnegative
integer entries bijectively to a pair (P, Q) of Young tableaux with common
shape.
1 2
2
Example of (semistandard) Young tableau T .
2 3
4
Rows weakly, columns strictly increasing.
4 4
5
Log-gamma polymer Warwick September 2011
λi = length of ith row
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Outline Introduction Burke property Tropical RSK
Combinatorial RSK (Robinson-Schensted-Knuth correspondence)
Classic RSK maps an n × N weight matrix Y = (Yi, j ) with nonnegative
integer entries bijectively to a pair (P, Q) of Young tableaux with common
shape.
1 2
2
Example of (semistandard) Young tableau T .
2 3
4
Rows weakly, columns strictly increasing.
4 4
5
λi = length of ith row
Shape sh(T ) = λ = (λ1 , . . . , λ4 ) = (3, 3, 2, 1).
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Combinatorial RSK (Robinson-Schensted-Knuth correspondence)
Classic RSK maps an n × N weight matrix Y = (Yi, j ) with nonnegative
integer entries bijectively to a pair (P, Q) of Young tableaux with common
shape.
1 2
2
Example of (semistandard) Young tableau T .
2 3
4
Rows weakly, columns strictly increasing.
4 4
5
λi = length of ith row
Shape sh(T ) = λ = (λ1 , . . . , λ4 ) = (3, 3, 2, 1).
x i, j = number of entries ≤ i in row j of the P-tableau.
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Combinatorial RSK (Robinson-Schensted-Knuth correspondence)
Classic RSK maps an n × N weight matrix Y = (Yi, j ) with nonnegative
integer entries bijectively to a pair (P, Q) of Young tableaux with common
shape.
1 2
2
Example of (semistandard) Young tableau T .
2 3
4
Rows weakly, columns strictly increasing.
4 4
5
λi = length of ith row
Shape sh(T ) = λ = (λ1 , . . . , λ4 ) = (3, 3, 2, 1).
x i, j = number of entries ≤ i in row j of the P-tableau.
x i, j ≤ x i−1, j−1 ≤ x i, j−1
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Outline Introduction Burke property Tropical RSK
{x i, j } can be arranged
in a Gelfand-Tsetlin pattern:
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Outline Introduction Burke property Tropical RSK
{x i, j } can be arranged
x 1,1
x 2,2
in a Gelfand-Tsetlin pattern:
...
x N,N
Log-gamma polymer Warwick September 2011
x 2,1
...
...
...
x N,2 x N,1
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Outline Introduction Burke property Tropical RSK
{x i, j } can be arranged
x 1,1
x 2,2
in a Gelfand-Tsetlin pattern:
Bottom row = shape.
Log-gamma polymer Warwick September 2011
...
x N,N
x 2,1
...
...
...
x N,2 x N,1
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Outline Introduction Burke property Tropical RSK
{x i, j } can be arranged
x 1,1
x 2,2
...
in a Gelfand-Tsetlin pattern:
x N,N
Bottom row = shape.
xk,1 + · · · + xk, ` =
max
π1 ,...,π` disjoint
x 2,1
...
...
...
x N,2 x N,1
X
Yi, j
(i, j) ∈ π1 ∪···∪ π`
where πm are up-right lattice paths in the weight matrix.
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Outline Introduction Burke property Tropical RSK
{x i, j } can be arranged
x 1,1
x 2,2
...
in a Gelfand-Tsetlin pattern:
x N,N
Bottom row = shape.
xk,1 + · · · + xk, ` =
x 2,1
...
...
...
x N,2 x N,1
X
max
π1 ,...,π` disjoint
Yi, j
(i, j) ∈ π1 ∪···∪ π`
where πm are up-right lattice paths in the weight matrix.
Compare this with tropical formula
zk,1 (n) · · · zk, ` (n) =
X
wt(π).
π∈Π`n,k
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Outline Introduction Burke property Tropical RSK
{x i, j } can be arranged
x 1,1
x 2,2
...
in a Gelfand-Tsetlin pattern:
x N,N
Bottom row = shape.
xk,1 + · · · + xk, ` =
x 2,1
...
...
...
x N,2 x N,1
X
max
π1 ,...,π` disjoint
Yi, j
(i, j) ∈ π1 ∪···∪ π`
where πm are up-right lattice paths in the weight matrix.
Compare this with tropical formula
zk,1 (n) · · · zk, ` (n) =
X
wt(π).
π∈Π`n,k
Difference is (+ , · ) vs. (max , +).
Log-gamma polymer Warwick September 2011
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Outline Introduction Burke property Tropical RSK
Consequences of Burke property: variance identity
Exit point of path from x-axis
ξx = max{k ≥ 0 : xi = (i, 0) for 0 ≤ i ≤ k}
ξx
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Outline Introduction Burke property Tropical RSK
Consequences of Burke property: variance identity
Exit point of path from x-axis
ξx = max{k ≥ 0 : xi = (i, 0) for 0 ≤ i ≤ k}
ξx
For θ, x > 0 define positive function
Z x
L(θ, x) =
Ψ0 (θ) − log y x −θ y θ−1 e x−y dy
0
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Outline Introduction Burke property Tropical RSK
Consequences of Burke property: variance identity
Exit point of path from x-axis
ξx = max{k ≥ 0 : xi = (i, 0) for 0 ≤ i ≤ k}
ξx
For θ, x > 0 define positive function
Z x
L(θ, x) =
Ψ0 (θ) − log y x −θ y θ−1 e x−y dy
0
Theorem. For the stationary case,
X
ξx
−1
Var log Zm,n = nΨ1 (µ − θ) − mΨ1 (θ) + 2 Em,n
L(θ, Yi,0 )
i=1
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Outline Introduction Burke property Tropical RSK
Variance identity leads to fluctuation bounds for log Z
With 0 < θ < µ fixed and N % ∞ assume
| m − NΨ1 (µ − θ) | ≤ CN 2/3
Log-gamma polymer Warwick September 2011
and | n − NΨ1 (θ) | ≤ CN 2/3
(1)
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Outline Introduction Burke property Tropical RSK
Variance identity leads to fluctuation bounds for log Z
With 0 < θ < µ fixed and N % ∞ assume
| m − NΨ1 (µ − θ) | ≤ CN 2/3
and | n − NΨ1 (θ) | ≤ CN 2/3
(1)
Theorem: Variance bounds along characteristic
For (m, n) as in (1),
C1 N 2/3 ≤ Var(log Zm,n ) ≤ C2 N 2/3 .
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Outline Introduction Burke property Tropical RSK
Variance identity leads to fluctuation bounds for log Z
With 0 < θ < µ fixed and N % ∞ assume
| m − NΨ1 (µ − θ) | ≤ CN 2/3
and | n − NΨ1 (θ) | ≤ CN 2/3
(1)
Theorem: Variance bounds along characteristic
For (m, n) as in (1),
C1 N 2/3 ≤ Var(log Zm,n ) ≤ C2 N 2/3 .
Theorem: Off-characteristic CLT
Suppose n = Ψ1 (θ)N and m = Ψ1 (µ − θ)N + γN α with γ > 0, α > 2/3.
Then
n
o
N −α/2 log Zm,n − E log Zm,n
⇒ N 0, γΨ1 (θ)
Log-gamma polymer Warwick September 2011
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