A lower-tail limit in the weak noise theory

TL;DR

This paper analyzes the variational problem related to the large deviation principle of the stochastic heat equation, confirming physics predictions about the behavior of solutions conditioned on unlikely events.

Contribution

It proves the convergence of the minimizer of the variational problem to an explicit function as the conditioned value approaches zero, confirming prior physics-based predictions.

Findings

Convergence of the minimizer to an explicit function as conditioning tends to zero

Validation of physics predictions regarding the lower-tail behavior

Explicit characterization of the most probable shape under conditioning

Abstract

We consider the variational problem associated with the Freidlin--Wentzell Large Deviation Principle of the Stochastic Heat Equation (SHE). The logarithm of the minimizer of the variational problem gives the most probable shape of the solution of the Kardar--Parisi--Zhang equation conditioned on achieving certain unlikely values. Taking the SHE with the delta initial condition and conditioning the value of its solution at the origin at a later time, under suitable scaling, we prove that the logarithm of the minimizer converges to an explicit function as we tune the value of the conditioning to $ 0 $. Our result confirms the physics prediction Kolokolov and Korshunov (2009), Meerson, Katzav, and Vilenkin (2016), Kamenev, Meerson, and Sasorov (2016).