Short Time Large Deviations of the KPZ Equation

Abstract

A. We consider the n-point, fixed-time large deviations of the KPZ equation with the narrow wedge initial condition. The scope consists of concave-configured, upper-tail deviations and a wide range of scaling regimes that allows time to be short, unit-order, and long. We prove the n-point large deviation principle and characterize, with proof, the corresponding spacetime limit shape. Our proof is based on the results -from the companion paper [Tsa23] -on moments of the stochastic heat equation and utilizes ideas coming from a tree decomposition. Behind our proof lies the phenomenon where the major contribution of the noise concentrates around certain corridors in spacetime, and we explicitly describe the corridors.

Upper-tail large deviation principle for the ASEP

“…In the short time regime, large deviations for the KPZ equation has been studied extensively in physics literature (see [49], [43], [42] and the references therein for a review). Recently, [55] rigorously derived the large deviation rate function of the KPZ equation in the short-time regime in a variational form and recovered deep lower-tail asymptotics, confirming existing physics predictions. For non-integrable models, large deviations of first-passage percolation were studied in [19] and more recently [5].…”

Upper-tail large deviation principle for the ASEP

“…In the short time regime, large deviations for the KPZ equation has been studied extensively in physics literature (see [LDMRS16], [KLD17], [Kra20] and the references therein for a review). Recently, [LT21] rigorously derived the large deviation rate function of the KPZ equation in the short-time regime in a variational form and recovered deep lower-tail asymptotics, confirming existing physics predictions. For non-integrable models, large deviations of first-passage percolation were studied in [CZ03] and more recently [BGS17].…”

Integrability in the weak noise theory

“…The second part of the main result, Theorem 2.3, gives explicit formulas for any minimizer in terms of its scattering coefficients. Together with the Freidlin-Wentzell LDP for the SHE proven in [LT21], the main result here gives a mathematically rigorous description of the LDP in terms of the NLS equations and the formulas.…”

“…The theory has seen much progress. Behaviors of the one-point rate function for various initial conditions and boundary conditions have been predicted [KK07, KK08, KK09, MKV16, KMS16, MS17, MV18, SM18, SMS18, ALM19, SMV19], some of which recently proven [LT21,GLLT21]; an intriguing symmetry breaking and second-order phase transition has been discovered in [JKM16,SKM18] via numerical means and analytically derived in [KLD17,KLD21b].…”