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 $\kappa>0$ the lower tail of $-\mathfrak{h}(\lfloor \kappa N \rfloor , N)$ satisfies a Large Deviation Principle (LDP) with speed $N^{2}$ and a rate function $\Phi _\kappa ^{(-)}$, which is given by the infimal deconvolution between a certain energy integral and a parabola. Our analysis begins with a distributional identity from [15], 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 [ 31], to convert the LDP for the shifted height function to the LDP for the stochastic six-vertex model height function.