Lower tail large deviations of the stochastic six vertex model

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 $\hat{I}\pm >0$ the lower tail of $-\mathfrak{h}(\lfloor \hat{I}\pm N\rfloor ,N)$ satisfies a Large Deviation Principle (LDP) with speed $N^2$ and a rate function $\hat{I}{¦}_{\hat{I}}{\pm }^{(-)}$, 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.

40 pages, 5 figures