Lower tail large deviations of the stochastic six vertex model

TL;DR

This paper establishes large deviation principles and log-concavity properties for the lower tail probabilities of the height function in the stochastic six-vertex model, using combinatorial, potential theory, and deconvolution techniques.

Contribution

It introduces a novel combinatorial approach and proves a large deviation principle for the lower tail of the height function, advancing understanding of its probabilistic behavior.

Findings

Lower tail probabilities are log-concave in a weak sense.

Established a large deviation principle with speed N^2 for the lower tail.

Derived a rate function characterized by an infimal deconvolution involving an energy integral.

Abstract

In this paper, we study lower tail probabilities of the height function $\mathfrak{h}(M,N)$ of the stochastic six-vertex model. We introduce a novel combinatorial approach to demonstrate that the tail probabilities $\mathbb{P}(\mathfrak{h}(M,N) \ge r)$ are log-concave in a certain weak sense. We prove further that for each $\alpha>0$ the lower tail of $-\mathfrak{h}(\lfloor \alpha N \rfloor, N)$ satisfies a Large Deviation Principle (LDP) with speed $N^2$ and a rate function $\Phi_\alpha^{(-)}$, which is given by the infimal deconvolution between a certain energy integral and a parabola. Our analysis begins with a distributional identity from BO17 [arXiv:1608.01564], which relates the lower tail of the height function, after a random shift, with a multiplicative functional of the Schur measure. Tools from potential theory allow us to extract the LDP for the shifted height function. We then use our weak log-concavity result, along with a deconvolution scheme from our earlier paper [arXiv:2307.01179], to convert the LDP for the shifted height function to the LDP for the stochastic six-vertex model height function.