Lower Tail Large Deviations of the Stochastic Six Vertex Model

In this paper, we study lower tail probabilities of the height function h(M,N) of the stochastic six-vertex model. We introduce a novel combinatorial approach to demonstrate that the tail probabilities P(h(M,N) ≥ r) are log-concave in a certain weak sense. We prove further that for each κ>0 the lower tail of -h( κ N , N) satisfies a Large Deviation Principle (LDP) with speed N² and a rate function Φ _κ ⁽⁻⁾, 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.