×

High temperature limits for \((1+1)\)-dimensional directed polymer with heavy-tailed disorder. (English) Zbl 1359.60117

The paper under review proves the conjecture that at the inverse temperature \(\beta n^{-\gamma}\) with \(\gamma = 1/4\) the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation under the assumption of six moments.
Let \(\Omega = \{\omega_v: v\in \mathbb Z^2\}\) be a collection of i.i.d. random variables indexed by vertices in \(\mathbb Z^2\). The set of nearest neighbor paths is given by \(\Psi_0^n=\{(i, s_i)_{0\leq i \leq n}: s_0=0, |s_i - s_{i-1}|=1, 1\leq i \leq n\}\). The energy of a path \({\mathbf s}=(i, s_i)_{0\leq i \leq n} \in \Phi_0^n\) is defined by \(H^{\omega} ({\mathbf s}) = \sum_{i=0}^n \omega_{i, s_i}\) and the polymer measure is given by \[ \frac{dP_{n, \beta_n}}{dP_n}({\mathbf s}) = (Z^{\omega}_{n, \beta_n})^{-1} \exp (\beta_n H^{\omega }({\mathbf s})), \] where \(Z^{\omega}_{n, \beta_n} = 2^{-n} \sum_{{\mathbf s} \in \Psi_0^n} \exp (\beta_n H^{\omega} ({\mathbf s}))\) and \(\beta_n >0\) is the inverse temperature that is allowed to depend on \(n\). The directed polymer model was introduced in [D. A. Huse and C. L. Henley, “Pinning and roughening of domain walls in Ising systems due to random impurities”, Phys. Rev. Lett. 54, No. 25, 2708–2711 (1985; doi:10.1103/PhysRevLett.54.2708)] for the interface of the two-dimensional Ising model with random interactions, and the directed polymer model is linked to the Kardar-Parisi-Zhang (KPZ) equation due to the fact that \(\log Z^{\omega}_{n, \beta_n}\) is the discretized solution of the Hopf-Cole solution to the KPZ equation. T. Alberts et al. [“Intermediate disorder regime for directed polymers in dimension \(1+1\)”, J. Phys. Rev. Lett. 105, No. 9, Article ID 090603, 4 p. (2010; doi:10.1103/PhysRevLett.105.090603); Ann. Probab. 42, No. 3, 1212–1256 (2014; Zbl 1292.82014)] have shown that under the assumption of exponential moments and \(\beta_n = \beta n^{-1/4}\), \[ \log Z^{\omega}_{n, \beta_n} - n \lambda (\beta_n) \to \log Z_{\sqrt{2}\beta} \text{ in distribution as}\;n\to \infty, \] where \(Z_{\sqrt{2}\beta}\) is the solution of the stochastic heat equation and \(\lambda (\beta_n)= \log E[e^{\beta_n \omega}]\). They conjectured that the same limit behavior holds under the assumption that the disorder possesses more than six moments. The main result of this paper is to prove this conjecture.
In Subsection 1.2, the authors describe the heuristic argument for the proof. First, split the plane \((\gamma, \alpha) \in (0, \infty)^2\) into seven different regions with different exponent behavior. The region \(R_2 \cup R_3 \cup R_4 \cup R_5\cup R_6\) is divided along families of curves characterized by the same exponent \(\xi \in [1/2, 1]\) determined by \(\xi = \frac{1+\alpha (1-\gamma)}{2\alpha -1}\) and \(\alpha \leq \frac{5-2\gamma}{1-\gamma}\) or \(\xi = \frac{2(1-\gamma)}{3}\) and \(\alpha \geq \frac{5-2\gamma}{1-\gamma}\); the region \(R_1\cup R_7\) has \(\xi = 1/2, \chi = 1/4 - \gamma < 2\xi -1\) on \(R_1\) and \(\xi=1, \chi = 2/\alpha - \gamma> 2\xi -1\) on \(R_7\). It is a classical result of order statistics that \[ \max\{\omega_v: v \in B_{n, \xi} = ([0, n]\times (-n^{\zeta}, n^{\zeta}))\cap \mathbb Z^2\} \sim n^{(1+\zeta)/\alpha}. \] The energy for the path \({\mathbf s}\) is \(\beta_nH^{\omega}({\mathbf s})\) and the fluctuation exponent \(\chi\) gives the typical order of \[ \log Z^{\omega}_{n, \beta_n} - E[\log Z^{\omega}_{n, \beta_n}] = \log \left\{ \exp (\beta_n (H^{\omega} - \lambda_{n, \beta_n}) )\right\} \sim n^{\chi + o(1)}. \] This contribution is negligible unless \[ 2\xi - 1 \leq (1+\xi)/\alpha - \gamma, \;\;\;\xi \leq(1+\alpha (1-\gamma))/(2\alpha -1), \] since \(n^{\chi + o(1) } \sim n^{ (1+\xi)/\alpha - \gamma} - n^{2\xi -1}\).
In Subsection 1.3, the authors explain the strategy behind the proofs of Theorem 1.1, 1.2 and 1.4, which lead to the proof of the conjecture. The authors show that the new environment behaves in a more regular way and prove a limit theorem for \(b_n (\log Z^{\tilde{\omega}}_{n, \beta_n} - a_n(k_n))\). The desired limit theorem for \(\log Z^{\tilde{\omega}}_{n, \beta_n}\) falls into the known case for \(\alpha \geq 6, b_n = 1, k_n=n^{1/4}\) as in Theorem 1.1. A similar strategy is applied in the case \(1/2< \alpha \leq 6\), with the cutoff chosen differently (\(k_n= m(n^{3/2}(\log n)^{\eta})/ m (n^{3/2})\) for \(\eta \in (1/2, \alpha )\)), leading to a more subtle multi-scale procedure with a logarithmic number of iterations. In Subsection 2.4, the behavior in region \(R_5\) is studied.
Section 2 starts with the well-known Feller result that the probability of the set \(B_j^c\) under the simple random walk path measure is bounded by \(4e^{-h_j^2/(2n)}\), where \(B_j = \{ ( i, s_i)_{0\leq i \leq n} \in \Psi_0^n: \max_{1\leq i \leq n} |s_i| < h_j\}\). The contribution of the set of paths in \(B_j \setminus B_{j-1}\) is small by Proposition 2.3.
The proof of Theorem 1.1 is given in Section 3, by Proposition 2.3, Proposition 3.2 and a Theorem of Alberts et al. [(2014), loc. cit.] that controls the limiting distribution of the partition function with truncated disorder converging to the desired quantity. Proposition 2.3 and Proposition 4.1 give a direct proof of Theorem 1.2 for the case of \(2< \alpha \leq 6, \gamma \geq 3/2 \alpha\) (Section 4). The proof of Theorem 1.4 is given in Section 5 with Proposition 2.3 and previous technical lemmas. The proofs of these technical Lemmas 1.3 and 1.6 are presented in the appendix.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
82D60 Statistical mechanics of polymers
60G57 Random measures
60G70 Extreme value theory; extremal stochastic processes

Citations:

Zbl 1292.82014
PDFBibTeX XMLCite
Full Text: DOI arXiv