Abstract
We consider directed polymer models involving multiple non-intersecting random walks moving through a space-time disordered environment in one spatial dimension. For a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling (in which time and space are scaled diffusively, and the strength of the environment is scaled to zero in a critical manner) the polymer partition function converges to the solution to the stochastic heat equation with multiplicative white noise. In this paper we prove the analogous result for multiple non-intersecting random walks started and ended grouped together. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren.
Highlights
Last passage percolation (LPP) involves finding the maximal sum of iid weights along random walk trajectories
Based on rigorous results for a few solvable choices of weights, it is widely conjectured that for generic weight distributions the fluctuations of this maximal sum grows like the cube-root of time, and has a limit under this scaling described by the Gaussian Unitary Ensemble (GUE) Tracy-Widom distribution
This distribution owes its name to the fact that it arises as the fluctuations of the largest eigenvalue of an N × N Gaussian Unitary Ensemble (GUE) matrix, as N goes to infinity [53]
Summary
Last passage percolation (LPP) involves finding the maximal sum of iid weights along random walk trajectories. As before, this is only proved for certain solvable models and even only in terms of the one-point marginal of the single path partition function – see e.g. The proof of Theorem 1.5 boils down to proving convergence of the correlation functions for nonintersecting random walks to those of non-intersecting Brownian motions This is the main technical result of this paper and is presented in Theorem 1.13. In [12], Corwin and Hammond considered a semi-discrete directed polymer model and showed that under the same intermediate disorder scaling considered here, the associated line ensemble is tight They defined any subsequential limit as a KPZ line ensemble and conjectured that there is a unique such limit which can be identified with O’Connell and Warren’s multi-layer extension. This naturally leads to the type of generalized multi-path polymers considered
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