Large Deviations for partition functions.
(...and for other polymer related quantities)
Nicos Georgiou
Department of Mathematics,
UW - Madison
Joint work with Timo Seppäläinen
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Layout
1
Inroduction and Results.
Quadrant directed polymers
The Results
Boundary log -gamma model
Burke Property
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Layout
1
Inroduction and Results.
Quadrant directed polymers
The Results
Boundary log -gamma model
Burke Property
2
The proofs.
Decomposition and Burke
Estimation
Thank you note !
Inversion
Proofs for unconstrained endpoint model
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Directed polymer in space-time random environment
Nearest neighbor, up-right path (x(u)),
u ∈ N2 .
“Space-time” environment
{ω(u) : u ∈ N2 }.
n
Π(m, n) = Up-right paths (xm,n ) from
(1, 1) to (m, n).
Quenched probability measure on the
paths
m
Qm,n {xm,n (·)} =
1
Zm,n
n
exp β
X
u∈xm,n (·)
Inverse temperature β > 0 (set to be 1 for the majority of the talk) .
The normalizing constant Zm,n is the partition function, given by
n X
o
X
Zm,n =
exp β
ω(u)
x∈Π(m,n)
u∈x(·)
P is the probability distribution on the environment ω, {ω(u)} i.i.d.
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
o
ω(u)
Questions and previous results
Question: Large deviations or Concentration Inequalities for the partition
function.
Concentration Inequalities:
1
2
3
Carmona - Hu: Order n concentration inequality (Gaussian
environment)
Comets - Shiga - Yoshida: Order n1/3 concentration inequality.
Liu - Watbled: Order n concentration inequality.
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Questions and previous results
Question: Large deviations or Concentration Inequalities for the partition
function.
Concentration Inequalities:
1
2
3
Carmona - Hu: Order n concentration inequality (Gaussian
environment)
Comets - Shiga - Yoshida: Order n1/3 concentration inequality.
Liu - Watbled: Order n concentration inequality.
Large deviations.
1
2
Carmona - Hu: Upper and lower tails normalizations (Gaussian
environment)
Ben-Ari: Lower tail large deviation regimes.
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Questions and previous results
Question: Large deviations or Concentration Inequalities for the partition
function.
Concentration Inequalities:
1
2
3
Carmona - Hu: Order n concentration inequality (Gaussian
environment)
Comets - Shiga - Yoshida: Order n1/3 concentration inequality.
Liu - Watbled: Order n concentration inequality.
Large deviations.
1
2
Carmona - Hu: Upper and lower tails normalizations (Gaussian
environment)
Ben-Ari: Lower tail large deviation regimes.
Explicit rate functions for partition functions ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Existence of Rate functions
Assumptions:
d-dimensional rectangle (only steps parallel to the positive axes are
allowed).
General i.i.d. weights, so that a ξ > 0 that depends on the
distributions of the weights ω exists, such that
E e ξ|ω(u)| < ∞.
β < ξ.
Theorem
For t > 0, u ∈ Rd+ and r ∈ R there exists a nonnegative function that
satisfies
β
Juβ (r ) = − lim n−1 log P{log Zbnuc
≥ nr }.
n→∞
J is convex in the variable (u, r ). The rate function is continuous in
(u, r ) where it is finite.
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Log-Gamma Model
Dimension 1 + 1
I.i.d. weights, with distributions
ω(i, j) ∼ log Yi,j ,
−1
where Yi,j
∼ Gamma(µ)
Gamma density: Γ(µ)−1 x µ−1 e −x
The partition function satisfies a law of large numbers:
lim n−1 log Zbnsc,bntc = fµ (s, t).
n→∞
Js,t (r ) = − lim n−1 log P{log Zbnsc,bntc ≥ nr }.
n→∞
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Results - The log-gamma rate function (fixed endpoint)
Definitions:
For 0 < ξ < (µ + ξ)/2 ≤ θ < µ, define
hξ (θ) = log Γ(µ − θ) − log Γ(µ − θ + ξ).
dξ (θ) = log Γ(θ − ξ) − log Γ(θ).
Theorem
Let r ∈ R, 0 ≤ s ≤ t. Then
Js,t (r ) = sup
sup
ξ∈[0,µ) hξ ((µ+ξ)/2)≤v
= sup
ξ∈[0,µ)
rξ −
(r , −s) · (ξ, v ) − t(dξ ◦ hξ−1 )(v ) ,
inf
{shξ (θ) + tdξ (θ)} ,
θ∈[(µ+ξ)/2,µ)
The function Js,t (r ) is strictly positive for r > r0 = fµ (s, t).
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Results - The log-gamma rate function (free endpoint)
Let s > 0. The free-endpoint directed polymer model has partition
function
n X
o
X
tot
=
exp
ω(u)
Zbnsc
x:bnsc-paths x
u∈x(·)
Theorem
Let r ∈ R, s > 0. Then
tot
Jstot (r ) = − lim n−1 log P{log Zbnsc
≥ nr }
n→∞
= Js/2,s/2 (r ).
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Results - Exit point and Path Chain Quenched LDP
Theorem (Exit point LDP)
Let 0 < t < 1 and consider (t, 1 − t) ∈ R2+ . For n ∈ N let
[n(t, 1 − t)] = (bntc, n − bntc) and denote by xn the last point of the
polymer chain x0,n . Then
fµ (1/2, 1/2) − fµ (t, 1 − t) = − lim n−1 log Qnω {xn = [n(t, 1 − t)]}.
n→∞
This readily leads to the following path large deviations.
Theorem (Polymer chain LDP)
Let γ(t) : [0, 1] 7→ R2+ be a non decreasing Lipschitz-1 curve with
γ(0) = 0 and let ε > 0 and Nε (γ) an ε-neighborhood of γ. Let
kγ(1)k1 = 1 . Then,
Z
fµ (1/2, 1/2) −
1
fµ (γ 0 (t)) dt = − lim lim n−1 log Qnω x0,n ∈ nNε (γ) .
ε→0 n→∞
0
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Computational tool: Log-gamma polymer with boundary
Define multiplicative weights Yi,j = e ω(i,j) independent.
Environment (distribution P)
Horizontal Weights U
−1
−1
Yi,0
= Ui,0
∼ Gamma(θ)
Vertical Weights V
−1
−1
Y0,j
= V0,j
∼ Gamma(µ − θ)
Gamma(µ) density:
Γ(µ)−1 x µ−1 e −x .
Bulk Weights Y
−1
Yi,j
∼ Gamma(µ)
This model allows specific calculations.
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Computational tool: Log-gamma polymer with boundary
Define multiplicative weights Yi,j = e ω(i,j) independent.
Environment (distribution P)
Horizontal Weights U
−1
−1
Yi,0
= Ui,0
∼ Gamma(θ)
Vertical Weights V
−1
−1
Y0,j
= V0,j
∼ Gamma(µ − θ)
Gamma(µ) density:
Γ(µ)−1 x µ−1 e −x .
Bulk Weights Y
−1
Yi,j
∼ Gamma(µ)
This model allows specific calculations. Reason: Burke property.
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Burke Property for the boundary model.
V0,j
Yi,j
Given initial weights (i, j ∈ N)
−1
Ui,0
∼ Gamma(θ)
V0,−1j ∼ Gamma(µ − θ)
0
1
Ui,0
Yi,−1
j ∼ Gamma(µ)
0
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Burke Property for the boundary model.
V0,j
Given initial weights (i, j ∈ N)
Yi,j
−1
Ui,0
∼ Gamma(θ)
V0,−1j ∼ Gamma(µ − θ)
0
1
Yi,−1
j ∼ Gamma(µ)
Ui,0
0
Compute Zm,n for all (m, n) ∈ Z2+ and then define
Um,n =
Zm,n
Zm−1,n
Vm,n =
Zm,n
Zm,n−1
Nicos Georgiou, UW -Madison
Xm,n =
Z
Zm,n −1
m,n
+
Zm+1,n
Zm,n+1
Large Deviations for partition functions.
Burke Property for the boundary model.
V0,j
Given initial weights (i, j ∈ N)
Yi,j
−1
Ui,0
∼ Gamma(θ)
V0,−1j ∼ Gamma(µ − θ)
0
1
Yi,−1
j ∼ Gamma(µ)
Ui,0
0
Compute Zm,n for all (m, n) ∈ Z2+ and then define
Um,n =
Zm,n
Zm−1,n
Vm,n =
Zm,n
Zm,n−1
(
For an undirected edge f :
Tf =
Nicos Georgiou, UW -Madison
Xm,n =
Ux
Vx
Z
Zm,n −1
m,n
+
Zm+1,n
Zm,n+1
f = {x − e1 , x}
f = {x − e2 , x}
Large Deviations for partition functions.
Burke Property
down-right path (zk ) with
edges
fk = {zk−1 , zk }, k ∈ Z
interior points I of path (zk )
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Burke Property
down-right path (zk ) with
edges
fk = {zk−1 , zk }, k ∈ Z
interior points I of path (zk )
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).
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Idea of Proof - Decomposition
We start by decomposing Zbnsc,bntc according to the exit point of the
polymer path from the boundary:
The shaded part represents
the partition function conditioned on (0, l) being the exit
point of the polymer path
from the boundary.
bntc
l
Y
l
1
0
0 1
bnsc
Nicos Georgiou, UW -Madison
V0,j Z(1,l)
(bnsc, bntc)
j=1
Large Deviations for partition functions.
Idea of Proof - Decomposition
Zbnsc,bntc =
X bnsc+bntc
Y
x·
=
Yxj
j=1
bntc l
X
Y
l=1
V0,j Z(1,l) (bnsc, bntc) +
j=1
bnsc k
X
Y
Ui,0 Z(k,1)
(bnsc, bntc) .
i=1
k=1
Consequence of the Burke Property:
bntc
Zbnsc,bntc =
Y
j=1
Nicos Georgiou, UW -Madison
bnsc
V0,j
Y
Ui,bntc
i=1
Large Deviations for partition functions.
Idea of Proof - Burke Trick
Divide both sides by
bnsc
Y
Ui,bntc =
i=1
Qbntc
j=1
bntc bntc
X
Y
l=1
V0,j :
−1
V0,j
Z(1,l)
(bnsc, bntc)
j=l+1
bnsc bntc
+
X
Y
k=1
j=1
−1
V0,j
k
Y
Ui,0
(bnsc, bntc)
Z(k,1)
i=1
bnsc
=
X
ηk Zk (bnsc, bntc).
k=−bntc
k6=0
Here we defined

bntc

 Y −1


V0,j ,



j=−k
ηk = η−1 ,


k

 Y


η
Ui,0 ,

0

for − bntc ≤ k ≤ −1,
k=0
for 0 < k ≤ bnsc,
i=1
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
.
Idea of Proof - Estimation
bnsc
(
Rs (r ) = − lim n
−1
n→∞
log P log
∼ −n
)
Ui,bntc ≥ nr
i=1
bnsc
(
−1
Y
log P log
)
X
ηk Zk (bnsc, bntc)
≥ nr
k=−bntc
k6=0
∼ −n−1 log P{max log ηk Zk (bnsc, bntc) ≥ nr }
k
∼ −n
∼
−1
inf
max log P{log ηk + log Zk (bnsc, bntc) ≥ nr }
k
a
inf {κa (x) + Js,t
(r − x)}
−t≤a≤s x∈R
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Thank You...
Questions & Comments ?
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Idea of Proof - Inversion
Rs∗ (ξ) =
sup
a ∗
κ∗a (ξ) + (Js,t
) (ξ)
a ∗
au(θ) − (−(Js,t
) (ξ))
−t≤a≤s
F s (θ, ξ) =
sup
−t≤a≤s
magic
F t (u −1 (v ), ξ) =
sup
0≤a≤t
t
F (u
−1
t
−1
(v ), ξ) = sup
a ∗
av − (−(Jt,t
) (ξ))
av − Gξ (a)
0≤a≤t
F (u
(v ), ξ) = Gξ∗ (v )
Then,
a
a ∗
Jt,t
(r ) = sup {r ξ − (Jt,t
) (ξ)}
ξ∈[0,µ)
= sup {r ξ + Gξ (a)}
ξ∈[0,µ)
= sup sup{r ξ + av − Gξ∗ (v )}.
ξ∈[0,µ) v ∈R
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.
Proof for unconstrained endpoint model
tot
log Zbn s c,bnsc−bn s c ≤ log Zbnsc
≤
2
2
≤ log(ns + 1) + log max Zk,bnsc−k .
k
bnsc
After some estimates, this translates
to the rate functions:
inf Ja,s−a (r ) ≤ Jstot (r ) ≤ Js/2,s/2 (r ).
bnsc − k
0≤a≤s
k bnsc
Use convexity of the point-to-point rate functions to get that
Js/2,s/2 (r ) ≤ inf Ja,s−a (r ).
0≤a≤s
Nicos Georgiou, UW -Madison
Large Deviations for partition functions.