Scaling exponents for certain 1+1 dimensional directed polymers

Scaling exponents for certain 1+1 dimensional directed polymers Timo Seppäläinen Department of Mathematics University of Wisconsin-Madison 2010 Scaling for a polymer 1/29

1 Introduction 2 KPZ equation 3 Log-gamma polymer Scaling for a polymer 2/29

Directed polymer in a random environment time N simple random walk path (x(t), t), t Z + 0 space Z d Scaling for a polymer 3/29

Directed polymer in a random environment time N simple random walk path (x(t), t), t Z + space-time environment {ω(x, t) : x Z d, t N} 0 space Z d Scaling for a polymer 3/29

Directed polymer in a random environment time N simple random walk path (x(t), t), t Z + space-time environment {ω(x, t) : x Z d, t N} inverse temperature β > 0 quenched probability measure on paths 0 space Z d Q n {x( )} = 1 { n } exp β ω(x(t), t) Z n partition function Z n = x( ) (summed over all n-paths) { exp β t=1 n } ω(x(t), t) t=1 Scaling for a polymer 3/29

General question: Behavior of model as β > 0 and dimension d vary. Especially whether x( ) is diffusive or not, that is, does it scale like standard RW. Scaling for a polymer 4/29

General question: Behavior of model as β > 0 and dimension d vary. Especially whether x( ) is diffusive or not, that is, does it scale like standard RW. Early results: diffusive behavior for d 3 and small β > 0: 1988 Imbrie and Spencer: n 1 E Q ( x(n) 2 ) c P-a.s. 1989 Bolthausen: quenched CLT for n 1/2 x(n). In the opposite direction: if d = 1, 2, or d 3 and β large enough, then c > 0 s.t. lim max n z Q n {x(n) = z} c P-a.s. (Comets, Shiga, Yoshida 2003) Scaling for a polymer 4/29

Definition of scaling exponents ζ and χ Fluctuations described by two scaling exponents: Scaling for a polymer 5/29

Definition of scaling exponents ζ and χ Fluctuations described by two scaling exponents: Fluctuations of the path {x(t) : 0 t n} are of order n ζ. Scaling for a polymer 5/29

Definition of scaling exponents ζ and χ Fluctuations described by two scaling exponents: Fluctuations of the path {x(t) : 0 t n} are of order n ζ. Fluctuations of log Z n are of order n χ. Conjecture for d = 1: ζ = 2/3 and χ = 1/3. Our results: these exact exponents for certain 1+1 dimensional models with particular weight distributions and boundary conditions. Scaling for a polymer 5/29

Earlier results for d = 1 exponents Past rigorous bounds give 3/5 ζ 3/4 and χ 1/8: Brownian motion in Poissonian potential: Wüthrich 1998, Comets and Yoshida 2005. Scaling for a polymer 6/29

Earlier results for d = 1 exponents Past rigorous bounds give 3/5 ζ 3/4 and χ 1/8: Brownian motion in Poissonian potential: Wüthrich 1998, Comets and Yoshida 2005. Gaussian RW in Gaussian potential: Petermann 2000 ζ 3/5, Mejane 2004 ζ 3/4 Scaling for a polymer 6/29

Models for which we can show ζ = 2/3 and χ = 1/3 (1) Log-gamma polymer: β = 1 and log ω(x, t) Gamma, plus appropriate boundary conditions Scaling for a polymer 7/29

Models for which we can show ζ = 2/3 and χ = 1/3 (1) Log-gamma polymer: β = 1 and log ω(x, t) Gamma, plus appropriate boundary conditions (2) Polymer in a Brownian environment. (Joint with B. Valkó.) Model introduced by O Connell and Yor 2001. (3) Continuum directed polymer, or Hopf-Cole solution of the Kardar-Parisi-Zhang (KPZ) equation with initial height function given by two-sided Brownian motion. (Joint with M. Balázs and J. Quastel.) Scaling for a polymer 7/29

Polymer in a Brownian environment Environment: independent Brownian motions B 1, B 2,..., B n Partition function (without boundary conditions): Z n,t (β) = 0<s 1 < <s n 1 <t exp [ β ( B 1 (s 1 ) + B 2 (s 2 ) B 2 (s 1 ) + + B 3 (s 3 ) B 3 (s 2 ) + + B n (t) B n (s n 1 ) )] ds 1,n 1 Scaling for a polymer 8/29

Hopf-Cole solution to KPZ equation KPZ eqn for height function h(t, x) of a 1+1 dim interface: h t = 1 2 h xx 1 2 (h x) 2 + Ẇ where Ẇ = Gaussian space-time white noise. Scaling for a polymer 9/29

Hopf-Cole solution to KPZ equation KPZ eqn for height function h(t, x) of a 1+1 dim interface: h t = 1 2 h xx 1 2 (h x) 2 + Ẇ where Ẇ = Gaussian space-time white noise. Initial height h(0, x) = two-sided Brownian motion for x R. Z = exp( h) satisfies Z t = 1 2 Z xx Z Ẇ that can be solved. Define h = log Z, the Hopf-Cole solution of KPZ. Scaling for a polymer 9/29

WASEP connection ζ ε (t, x) height process of weakly asymmetric simple exclusion s.t. ζ ε (x + 1) ζ ε (x) = ±1 Scaling for a polymer 10/29

WASEP connection ζ ε (t, x) height process of weakly asymmetric simple exclusion s.t. ζ ε (x + 1) ζ ε (x) = ±1 rate up 1 2 + ε rate down 1 2 Scaling for a polymer 10/29

WASEP connection Jumps: ζ ε (x) { ζ ε (x) + 2 with rate 1/2 + ε if ζ ε (x) is a local min ζ ε (x) 2 with rate 1/2 if ζ ε (x) is a local max Scaling for a polymer 11/29

WASEP connection Jumps: ζ ε (x) { ζ ε (x) + 2 with rate 1/2 + ε if ζ ε (x) is a local min ζ ε (x) 2 with rate 1/2 if ζ ε (x) is a local max Initially: ζ ε (0, x + 1) ζ ε (0, x) = ±1 with probab 1 2. Scaling for a polymer 11/29

WASEP connection Jumps: ζ ε (x) { ζ ε (x) + 2 with rate 1/2 + ε if ζ ε (x) is a local min ζ ε (x) 2 with rate 1/2 if ζ ε (x) is a local max Initially: ζ ε (0, x + 1) ζ ε (0, x) = ±1 with probab 1 2. h ε (t, x) = ε 1/2 ( ζ ε (ε 2 t, [ε 1 x]) v ε t ) Scaling for a polymer 11/29

Fluctuation bounds From coupling arguments for WASEP C 1 t 2/3 Var(h ε (t, 0)) C 2 t 2/3 Scaling for a polymer 12/29

Fluctuation bounds From coupling arguments for WASEP C 1 t 2/3 Var(h ε (t, 0)) C 2 t 2/3 Thm. (Balázs-Quastel-S) For the Hopf-Cole solution of KPZ, C 1 t 2/3 Var(h(t, 0)) C 2 t 2/3 Scaling for a polymer 12/29

Fluctuation bounds From coupling arguments for WASEP C 1 t 2/3 Var(h ε (t, 0)) C 2 t 2/3 Thm. (Balázs-Quastel-S) For the Hopf-Cole solution of KPZ, C 1 t 2/3 Var(h(t, 0)) C 2 t 2/3 The lower bound comes from control of rescaled correlations S ε (t, x) = ε 1 Cov [ η(ε 2 t, ε 1 x), η(0, 0) ] Scaling for a polymer 12/29

Rescaled correlations: S ε (t, x) = ε 1 Cov [ η(ε 2 t, ε 1 x), η(0, 0) ] Scaling for a polymer 13/29

Rescaled correlations: S ε (t, x) = ε 1 Cov [ η(ε 2 t, ε 1 x), η(0, 0) ] S ε (t, x)dx S(t, dx) with control of moments: x m S ε (t, x) dx O(t 2m/3 ), 1 m < 3. (A second class particle estimate.) Scaling for a polymer 13/29

Rescaled correlations: S ε (t, x) = ε 1 Cov [ η(ε 2 t, ε 1 x), η(0, 0) ] S ε (t, x)dx S(t, dx) with control of moments: x m S ε (t, x) dx O(t 2m/3 ), 1 m < 3. (A second class particle estimate.) S(t, dx) = 1 2 xx Var(h(t, x)) as distributions. Scaling for a polymer 13/29

Polymer in first quadrant with fixed endpoints n 0 0 m We turn the picture 45 degrees. Polymer is an up-right path from (0, 0) to (m, n) in Z 2 +. Scaling for a polymer 14/29

Notation Π m,n = { up-right paths (x k ) from (0, 0) to (m, n) } Scaling for a polymer 15/29

Notation Π m,n = { up-right paths (x k ) from (0, 0) to (m, n) } Fix β = 1. Y i, j = e ω(i,j) independent. Environment (Y i, j : (i, j) Z 2 +) with distribution P. Scaling for a polymer 15/29

Notation Π m,n = { up-right paths (x k ) from (0, 0) to (m, n) } Fix β = 1. Y i, j = e ω(i,j) independent. Environment (Y i, j : (i, j) Z 2 +) with distribution P. Z m,n = x Π m,n m+n k=1 Y xk Q m,n (x ) = 1 Z m,n m+n k=1 Y xk Scaling for a polymer 15/29

Weight distributions For i, j N: V 0, j Y i, j U 1 i,0 Gamma(θ) V 1 0, j Gamma(µ θ) 0 1 0 U i,0 Y 1 i, j Gamma(µ) parameters 0 < θ < µ Gamma(θ) density: Γ(θ) 1 x θ 1 e x on R + Scaling for a polymer 16/29

Weight distributions For i, j N: V 0, j Y i, j U 1 i,0 Gamma(θ) V 1 0, j Gamma(µ θ) 0 1 0 U i,0 Y 1 i, j Gamma(µ) parameters 0 < θ < µ Gamma(θ) density: Γ(θ) 1 x θ 1 e x on R + E(log U) = Ψ 0 (θ) and Var(log U) = Ψ 1 (θ) Ψ n (s) = (d n+1 /ds n+1 ) log Γ(s) Scaling for a polymer 16/29

Variance bounds for log Z With 0 < θ < µ fixed and N assume m NΨ 1 (µ θ) CN 2/3 and n NΨ 1 (θ) CN 2/3 (1) Scaling for a polymer 17/29

Variance bounds for log Z With 0 < θ < µ fixed and N assume m NΨ 1 (µ θ) CN 2/3 and n NΨ 1 (θ) CN 2/3 (1) Theorem For (m, n) as in (1), C 1 N 2/3 Var(log Z m,n ) C 2 N 2/3. Scaling for a polymer 17/29

Variance bounds for log Z With 0 < θ < µ fixed and N assume m NΨ 1 (µ θ) CN 2/3 and n NΨ 1 (θ) CN 2/3 (1) Theorem For (m, n) as in (1), C 1 N 2/3 Var(log Z m,n ) C 2 N 2/3. Theorem Suppose n = Ψ 1 (θ)n and m = Ψ 1 (µ θ)n + γn α with γ > 0, α > 2/3. Then N α/2{ log Z m,n E ( log Z m,n ) } N ( 0, γψ 1 (θ) ) Scaling for a polymer 17/29

Fluctuation bounds for path v 0 (j) = leftmost, v 1 (j) = rightmost point of x on horizontal line: v 0 (j) = min{i {0,..., m} : k : x k = (i, j)} v 1 (j) = max{i {0,..., m} : k : x k = (i, j)} Scaling for a polymer 18/29

Fluctuation bounds for path v 0 (j) = leftmost, v 1 (j) = rightmost point of x on horizontal line: v 0 (j) = min{i {0,..., m} : k : x k = (i, j)} v 1 (j) = max{i {0,..., m} : k : x k = (i, j)} Theorem Assume (m, n) as previously and 0 < τ < 1. Then { (a) P v 0 ( τn ) < τm bn 2/3 or v 1 ( τn ) > τm + bn 2/3} C b 3 (b) ε > 0 δ > 0 such that lim P{ k such that x k (τm, τn) δn 2/3 } ε. N Scaling for a polymer 18/29

Results for log-gamma polymer summarized With reciprocals of gammas for weights, both endpoints of the polymer fixed and the right boundary conditions on the axes, we have identified the one-dimensional exponents ζ = 2/3 and χ = 1/3. Next step is to eliminate the boundary conditions and consider polymers with fixed length and free endpoint Scaling for a polymer 19/29

Burke property for log-gamma polymer with boundary Given initial weights (i, j N): V 0, j Y i, j U 1 i,0 Gamma(θ), V 1 0, j Gamma(µ θ) 0 1 0 U i,0 Y 1 i, j Gamma(µ) Scaling for a polymer 20/29

Burke property for log-gamma polymer with boundary Given initial weights (i, j N): V 0, j Y i, j U 1 i,0 Gamma(θ), V 1 0, j Gamma(µ θ) 0 1 0 U i,0 Y 1 i, j Gamma(µ) Compute Z m,n for all (m, n) Z 2 + and then define U m,n = Z m,n V m,n = Z ( m,n Zm,n X m,n = + Z ) m,n 1 Z m 1,n Z m,n 1 Z m+1,n Z m,n+1 Scaling for a polymer 20/29

down-right path (z k ) with edges f k = {z k 1, z k }, k Z interior points I of path (z k ) Scaling for a polymer 21/29

down-right path (z k ) with edges f k = {z k 1, z k }, k Z interior points I of path (z k ) Theorem Variables {T fk, X z : 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 is a generalization of Burke s Theorem for M/M/1 queues, via the last-passage representation of M/M/1 queues in series. Scaling for a polymer 21/29

Proof of Burke property Induction on I by flipping a growth corner: V Y U U U = Y (1 + U/V ) V = Y (1 + V /U) X V X = ( U 1 + V 1) 1 Scaling for a polymer 22/29

Proof of Burke property Induction on I by flipping a growth corner: V Y U U U = Y (1 + U/V ) V = Y (1 + V /U) X V X = ( U 1 + V 1) 1 Lemma. Given that (U, V, Y ) are independent positive r.v. s, (U, V, X ) = d (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). Scaling for a polymer 22/29

Proof of off-characteristic CLT Recall that n = Ψ 1 (θ)n γ > 0, α > 2/3. m = Ψ 1 (µ θ)n + γn α Scaling for a polymer 23/29

Proof of off-characteristic CLT Recall that n = Ψ 1 (θ)n γ > 0, α > 2/3. m = Ψ 1 (µ θ)n + γn α Set m 1 = Ψ 1 (µ θ)n. Scaling for a polymer 23/29

Proof of off-characteristic CLT Recall that n = Ψ 1 (θ)n γ > 0, α > 2/3. m = Ψ 1 (µ θ)n + γn α Set m 1 = Ψ 1 (µ θ)n. Since Z m,n = Z m1,n m i=m 1 +1 U i,n Scaling for a polymer 23/29

Proof of off-characteristic CLT Recall that n = Ψ 1 (θ)n γ > 0, α > 2/3. m = Ψ 1 (µ θ)n + γn α Set m 1 = Ψ 1 (µ θ)n. Since Z m,n = Z m1,n m i=m 1 +1 U i,n N α/2 log Z m,n = N α/2 log Z m1,n + N α/2 m i=m 1 +1 log U i,n First term on the right is O(N 1/3 α/2 ) 0. Second term is a sum of order N α i.i.d. terms. Scaling for a polymer 23/29

Variance identity ξ x Exit point of path from x-axis ξ x = max{k 0 : x i = (i, 0) for 0 i k} Scaling for a polymer 24/29

Variance identity ξ x Exit point of path from x-axis ξ x = max{k 0 : x i = (i, 0) for 0 i k} For θ, x > 0 define positive function L(θ, x) = x 0 ( Ψ0 (θ) log y ) x θ y θ 1 e x y dy Scaling for a polymer 24/29

Variance identity ξ x Exit point of path from x-axis ξ x = max{k 0 : x i = (i, 0) for 0 i k} For θ, x > 0 define positive function Theorem. L(θ, x) = x 0 ( Ψ0 (θ) log y ) x θ y θ 1 e x y dy For the model with boundary, Var [ log Z m,n ] = nψ1 (µ θ) mψ 1 (θ) + 2 E m,n [ ξx i=1 ] L(θ, Y 1 i,0 ) Scaling for a polymer 24/29

Variance identity, sketch of proof N = log Z m,n log Z 0,n W = log Z 0,n E = log Z m,n log Z m,0 S = log Z m,0 Scaling for a polymer 25/29

Variance identity, sketch of proof N = log Z m,n log Z 0,n W = log Z 0,n E = log Z m,n log Z m,0 S = log Z m,0 Var [ ] log Z m,n = Var(W + N) = Var(W ) + Var(N) + 2 Cov(W, N) = Var(W ) + Var(N) + 2 Cov(S + E N, N) = Var(W ) Var(N) + 2 Cov(S, N) (E, N ind.) = nψ 1 (µ θ) mψ 1 (θ) + 2 Cov(S, N). Scaling for a polymer 25/29

To differentiate w.r.t. parameter θ of S while keeping W fixed, introduce a separate parameter ρ (= µ θ) for W. Cov(S, N) = θ E(N) Scaling for a polymer 26/29

To differentiate w.r.t. parameter θ of S while keeping W fixed, introduce a separate parameter ρ (= µ θ) for W. Cov(S, N) = Ẽ[ ] θ E(N) = θ log Z m,n(θ) Scaling for a polymer 26/29

To differentiate w.r.t. parameter θ of S while keeping W fixed, introduce a separate parameter ρ (= µ θ) for W. Cov(S, N) = Ẽ[ ] θ E(N) = θ log Z m,n(θ) when Z m,n (θ) = x Π m,n ξ x i=1 H θ (η i ) 1 m+n k=ξ x +1 Y xk with η i IID Unif(0, 1), x H θ (η) = F 1 y θ 1 e y θ (η), F θ (x) = dy. 0 Γ(θ) Scaling for a polymer 26/29

Together: Var [ log Z m,n ] = nψ1 (µ θ) mψ 1 (θ) + 2 Cov(S, N) = nψ 1 (µ θ) mψ 1 (θ) + 2 E m,n [ ξx i=1 L(θ, Y 1 i,0 ) ]. This was the claimed formula. Scaling for a polymer 27/29

Sketch of upper bound proof The argument develops an inequality that controls both log Z and ξ x simultaneously. Introduce an auxiliary parameter λ = θ bu/n. Scaling for a polymer 28/29

Sketch of upper bound proof The argument develops an inequality that controls both log Z and ξ x simultaneously. Introduce an auxiliary parameter λ = θ bu/n. The weight of a path x such that ξ x > 0 satisfies W (θ) = ξ x i=1 H θ (η i ) 1 m+n k=ξ x +1 Y xk = W (λ) ξ x i=1 H λ (η i ) H θ (η i ). Scaling for a polymer 28/29

Sketch of upper bound proof The argument develops an inequality that controls both log Z and ξ x simultaneously. Introduce an auxiliary parameter λ = θ bu/n. The weight of a path x such that ξ x > 0 satisfies W (θ) = ξ x i=1 Since H λ (η) H θ (η), H θ (η i ) 1 Q θ,ω {ξ x u} = 1 Z(θ) x m+n k=ξ x +1 Y xk = W (λ) ξ x i=1 H λ (η i ) H θ (η i ). 1{ξ x u}w (θ) Z(λ) u Z(θ) H λ (η i ) H θ (η i ). i=1 Scaling for a polymer 28/29

For 1 u δn and 0 < s < δ, P [ Q ω {ξ x u} e su2 /N ] { u } H λ (η i ) P H θ (η i ) α i=1 ( Z(λ) + P Z(θ) α 1 e su2 /N ). Scaling for a polymer 29/29

For 1 u δn and 0 < s < δ, P [ Q ω {ξ x u} e su2 /N ] { u } H λ (η i ) P H θ (η i ) α i=1 ( Z(λ) + P Z(θ) α 1 e su2 /N Choose α right. Bound these probabilities with Chebychev which brings Var(log Z) into play. In the characteristic rectangle Var(log Z) can be bounded by E(ξ x ). The end result is this inequality: P [ Q ω {ξ x u} e su2 /N ] CN2 u 4 E(ξ x) + CN2 u 3 ). Scaling for a polymer 29/29