Large deviations and fluctuation exponents for some polymer models. Directed polymer in a random environment. KPZ equation Log-gamma polymer

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1 Large deviations and fluctuation exponents for some polymer models Timo Seppäläinen Department of Mathematics University of Wisconsin-Madison Introduction 2 Large deviations 3 Fluctuation exponents KPZ equation Log-gamma polymer Polymer large deviations and fluctuations SEPC May 16, 211 1/35 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 > quenched probability measure on paths 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 n t=1 } ω(x(t, t P probability distribution on ω, often ω(x, t} i.i.d. t=1 Polymer large deviations and fluctuations SEPC May 16, 211 2/35 Key quantities again: Quenched measure Q n x( } = Z 1 n Partition function Z n = x( Questions: exp β exp β n t=1 n } ω(x(t, t t=1 } ω(x(t, t Behavior of walk x( under Q n on large scales: fluctuation exponents, central limit theorems, large deviations Behavior of log Z n (now also random as a function of ω Dependence on β and d Polymer large deviations and fluctuations SEPC May 16, 211 3/35 Polymer large deviations and fluctuations SEPC May 16, 211 4/35

2 Large deviations Question: describe quenched limit lim n n 1 log Z n Large deviation perspective. (P-a.s. Generalize: E = expectation under background RW X n on Z ν. n 1 log Z n = n 1 log E e β P n 1 k= ω X k = n 1 log E e P n 1 k= g(ω X k = n 1 log E e P n 1 k= g(t X k ω, Z k+1,k+l Introduced shift (T x ω y = ω x+y, steps Z k = X k X k 1 R, Z 1,l = (Z 1, Z 2,..., Z l. g(ω, z 1,l is a function on Ω l = Ω R l. n 1 Define empirical measure R n = n 1 δ TXk ω,z k+1,k+l. It is a probability measure on Ω l. k= Then n 1 log Z n = n 1 log E e nr n (g Task: understand large deviations of P R n } under P-a.e. fixed ω (quenched. Process: Markov chain (T Xn ω, Z n+1,n+l on Ω l under a fixed ω. Evolution: pick random step z from R, then execute move S z : (ω, z 1,l (T z1 ω, z 2,l z. Defines kernel p on Ω l : p(η, S z η = R 1. Polymer large deviations and fluctuations SEPC May 16, 211 5/35 Entropy For µ M 1 (Ω l, q Markov kernel on Ω l, usual relative entropy on Ω 2 l : H(µ q µ p = Ω l z R The effect of P in the background? q(η, S z η log q(η, S zη p(η, S z η µ(dη. Polymer large deviations and fluctuations SEPC May 16, 211 6/35 Assumptions. Environment ω x } IID under P. g local function on Ω l, E g p < for some p > ν. Theorem. (Rassoul-Agha, S, Yilmaz Deterministic limit Λ(g = lim n n 1 log E e nr n (g exists P-a.s. Let µ = Ω-marginal of µ M 1 (Ω l. Define inf H(µ q µ p : µq = µ } H P (µ = Infimum taken over Markov kernels q that fix µ. H P is convex but not lower semicontinuous. if µ P otherwise. and Remarks. Λ(g = H # P (g sup µ sup E µ g c H P (µ }. c> With higher moments of g admit mixing P. Λ(g >. IID directed + above moment Λ(g finite. Polymer large deviations and fluctuations SEPC May 16, 211 7/35 Polymer large deviations and fluctuations SEPC May 16, 211 8/35

3 Quenched weak LDP (large deviation principle under Q n. Rate function Q n (A = 1 E e nr n (g E e nr n (g 1 A (ω, Z 1, I (µ = inf c> H P(µ E µ (g c + Λ(g }. Theorem. (RSY Assumptions as above and Λ(g finite. Then P-a.s. for compact F M 1 (Ω l and open G M 1 (Ω l : lim n n 1 log Q n R n F } inf I (µ µ F lim n 1 log Q n R n G} inf I (µ n µ G IID environment, directed walk: full LDP holds. Return to d + 1 dim directed polymer in i.i.d. environment. Question: Is the path x( diffusive or not, that is, does it scale like standard RW? Early results: diffusive behavior for d 3 and small β > : 1988 Imbrie and Spencer: n 1 E Q ( x(n 2 c P-a.s 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 > s.t. lim max Q n x(n = z} c P-a.s. n z (Carmona and Hu 22, Comets, Shiga, and Yoshida 23 Polymer large deviations and fluctuations SEPC May 16, 211 9/35 Definition of fluctuation exponents ζ and χ Polymer large deviations and fluctuations SEPC May 16, 211 1/35 Earlier results for d = 1 exponents Fluctuations of the path x(t : 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. Results: these exact exponents for three particular 1+1 dimensional models. Past rigorous bounds give 3/5 ζ 3/4 and χ 1/8: Brownian motion in Poissonian potential: Wüthrich 1998, Comets and Yoshida 25. Gaussian RW in Gaussian potential: Petermann 2 ζ 3/5, Mejane 24 ζ 3/4 Licea, Newman, Piza : corresponding results for first passage percolation Polymer large deviations and fluctuations SEPC May 16, /35 Polymer large deviations and fluctuations SEPC May 16, /35

4 Rigorous ζ = 2/3 and χ = 1/3 results exist for three exactly solvable models: (1 Log-gamma polymer: β = 1 and e 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 21. (3 Continuum directed polymer, or Hopf-Cole solution of the Kardar-Parisi-Zhang (KPZ equation: (i Initial height function given by two-sided Brownian motion (joint with M. Balázs and J. Quastel. (ii Narrow wedge initial condition (Amir, Corwin, Quastel. Next details on (3.i, then details on (1. Polymer large deviations and fluctuations SEPC May 16, /35 WASEP connection 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(, 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. Bertini-Giacomin (1997: h can be obtained as a weak limit via a smoothing and renormalization of KPZ. Polymer large deviations and fluctuations SEPC May 16, /35 WASEP connection ζ ε (t, x height process of weakly asymmetric simple exclusion s.t. ζ ε (x + 1 ζ ε (x = ±1 Jumps: ζε (x + 2 with rate 1 2 ζ ε (x + 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 rate up ε Initially: ζ ε (, x + 1 ζ ε (, x = ±1 with probab 1 2. rate down 1 2 h ε (t, x = ε 1/2 ( ζ ε (ε 2 t, ε 1 x v ε t Theorem (Bertini-Giacomin 1997 As ε, h ε h Polymer large deviations and fluctuations SEPC May 16, /35 Polymer large deviations and fluctuations SEPC May 16, /35

5 Fluctuation bounds From coupling arguments for WASEP C 1 t 2/3 Var(h ε (t, C 2 t 2/3 Theorem (Balázs-Quastel-S For the Hopf-Cole solution of KPZ, C 1 t 2/3 Var(h(t, C 2 t 2/3 Lower bound comes from control of rescaled correlations S ε (t, x = 4ε 1 Cov η(ε 2 t, ε 1 x, η(, where η(t, x, 1} is the occupation variable of WASEP Polymer large deviations and fluctuations SEPC May 16, /35 Rescaled correlations again: S ε (t, x = 4ε 1 Cov η(ε 2 t, ε 1 x, η(, E ϕ, h ε (t ψ, h ε ( = 1 2 ( y + x ϕ 2 ψ ( y x 2 dy Let ε. On the left increments of h ε so total control! S ε (t, x dx On the right S ε (t, xdx S(t, dx with control of moments: x m S ε (t, x dx O(t 2m/3, 1 m < 3. (Second class particle estimate. Polymer large deviations and fluctuations SEPC May 16, / dimensional lattice polymer with log-gamma weights After ε limit E ϕ, h(t ψ, h( = 1 2 From mean zero, stationary h increments ( y + x ϕ 2 ψ ( y x 1 2 xx Var(h(t, x = S(t, dx as distributions. With some control over tails we arrive at the result: Var(h(t, = x S(t, dx O(t 2/3. 2 dy S(t, dx Fix both endpoints. n m Π m,n = set of admissible paths independent weights Y i, j = e ω(i,j environment (Y i, j : (i, j Z 2 + Z m,n = x quenched measure Q m,n (x = Z 1 m,n m+n k=1 m+n k=1 Y xk Y xk averaged measure P m,n (x = EQ m,n (x Polymer large deviations and fluctuations SEPC May 16, /35 Polymer large deviations and fluctuations SEPC May 16, 211 2/35

6 Weight distributions Variance bounds for log Z Parameters < θ < µ. Bulk weights Y i, j for i, j N Boundary weights U i, = Y i, and V, j = Y, j. V, j 1 Y i, j U i, U 1 i, Gamma(θ V 1, j Gamma(µ θ Y 1 i, j Gamma(µ Gamma(θ density: Γ(θ 1 x θ 1 e x on R + Ψ n (s = (d n+1 /ds n+1 log Γ(s E(log U = Ψ (θ and Var(log U = Ψ 1 (θ With < θ < µ fixed and N assume Theorem m NΨ 1 (µ θ CN 2/3 and n NΨ 1 (θ CN 2/3 (1 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 (θn and m = Ψ 1 (µ θn + γn α with γ >, α > 2/3. Then N α/2 log Z m,n E ( log Z m,n } N (, γψ 1 (θ Polymer large deviations and fluctuations SEPC May 16, /35 Polymer large deviations and fluctuations SEPC May 16, /35 Fluctuation bounds for path v (j = leftmost, v 1 (j = rightmost point of x on horizontal line: v (j = mini,..., m} : k : x k = (i, j} v 1 (j = maxi,..., m} : k : x k = (i, j} 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. Theorem Assume (m, n as previously and < τ < 1. Then (a P v ( τn < τm bn 2/3 or v 1 ( τn > τm + bn 2/3} C b 3 (b ε > δ > such that lim P k such that x k (τm, τn δn 2/3 } ε. N Next step is to eliminate the boundary conditions and consider polymers with fixed length and free endpoint In both scenarios we have the upper bounds for both log Z and the path. But currently do not have the lower bounds. Next some key points of the proof. Polymer large deviations and fluctuations SEPC May 16, /35 Polymer large deviations and fluctuations SEPC May 16, /35

7 Burke property for log-gamma polymer with boundary V, j 1 Y i, j U i, Given initial weights (i, j N: U 1 i, Gamma(θ, V 1, j Gamma(µ θ 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 For an undirected edge f : T f = U x f = x e 1, x} V x f = x e 2, x} Theorem down-right path (z k with edges f k = z k 1, z k }, k Z interior points I of path (z k 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. Polymer large deviations and fluctuations SEPC May 16, /35 Proof of Burke property Polymer large deviations and fluctuations SEPC May 16, /35 Proof of off-characteristic CLT Induction on I by flipping a growth corner: V Lemma. Y U U U = Y (1 + U/V V = Y (1 + V /U X V X = ( U 1 + V 1 1 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. This gives all (z k with finite I. General case follows. Recall that Set m 1 = Ψ 1 (µ θn. n = Ψ 1 (θn γ >, α > 2/3. m = Ψ 1 (µ θn + γn α 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. Second term is a sum of order N α i.i.d. terms. Polymer large deviations and fluctuations SEPC May 16, /35 Polymer large deviations and fluctuations SEPC May 16, /35

8 Variance identity Variance identity, sketch of proof ξ x Exit point of path from x-axis ξ x = maxk : x i = (i, for i k} N = log Z m,n log Z,n W = log Z,n E = log Z m,n log Z m, S = log Z m, For θ, x > define positive function Theorem. L(θ, x = x For the model with boundary, ( Ψ (θ log y x θ y θ 1 e x y dy Var log Z m,n = nψ1 (µ θ mψ 1 (θ + 2 E m,n ξx L(θ, Y 1 i, 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. Polymer large deviations and fluctuations SEPC May 16, /35 Polymer large deviations and fluctuations SEPC May 16, 211 3/35 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 Together: when Z m,n (θ = x Π m,n η i IID Unif(, 1, ξ x H θ (η i 1 m+n k=ξ x +1 Y xk with x H θ (η = F 1 θ (η, F θ (x = y θ 1 e y Γ(θ dy. 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 This was the claimed formula. L(θ, Y 1 i,. Differentiate: θ log Z ξx m,n(θ = E Q m,n L(θ, Y 1 i,. Polymer large deviations and fluctuations SEPC May 16, /35 Polymer large deviations and fluctuations SEPC May 16, /35

9 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 > satisfies W (θ = ξ x 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 H λ (η i H θ (η i. 1ξ x u}w (θ Z(λ u Z(θ H λ (η i H θ (η i. For 1 u δn and < s < δ, P Q ω ξ x u} e su2 /N u } H λ (η i P H θ (η i α ( 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 Handle u δn with large deviation estimates. In the end, integration gives the moment bounds. END.. Polymer large deviations and fluctuations SEPC May 16, /35 Polymer large deviations and fluctuations SEPC May 16, /35 Polymer in a Brownian environment Environment: independent Brownian motions B 1, B 2,..., B n Partition function (without boundary conditions: Z n,t (β = <s 1 < <s n 1 <t exp β ( B 1 (s 1 + B 2 (s 2 B 2 (s B 3 (s 3 B 3 (s B n (t B n (s n 1 ds 1,n 1 Polymer large deviations and fluctuations SEPC May 16, /35