Symmetries and Zero Modes in Sample Path Large Deviations

Abstract

Sharp large deviation estimates for stochastic differential equations with small noise, based on minimizing the Freidlin–Wentzell action functional under appropriate boundary conditions, can be obtained by integrating certain matrix Riccati differential equations along the large deviation minimizers or instantons, either forward or backward in time. Previous works in this direction often rely on the existence of isolated minimizers with positive definite second variation. By adopting techniques from field theory and explicitly evaluating the large deviation prefactors as functional determinant ratios using Forman’s theorem, we extend the approach to general systems where degenerate submanifolds of minimizers exist. The key technique for this is a boundary-type regularization of the second variation operator. This extension is particularly relevant if the system possesses continuous symmetries that are broken by the instantons. We find that removing the vanishing eigenvalues associated with the zero modes is possible within the Riccati formulation and amounts to modifying the initial or final conditions and evaluation of the Riccati matrices. We apply our results in multiple examples including a dynamical phase transition for the average surface height in short-time large deviations of the one-dimensional Kardar–Parisi–Zhang equation with flat initial profile.

Short-time large deviations of the spatially averaged height of a Kardar–Parisi–Zhang interface on a ring

“…For small ℓ a truly second-order DPT is observed as predicted earlier [10,22]. At intermediate values of ℓ = O(1) DPTs of both types occur.…”

“…Both solutions are one-soliton solutions for branches 2 and 3, respectively. The action of the solution in the top row is S = 28 723.766, which deviates by 0.4% from the predicted large-ℓ result S = 28 590.660 given by equation (50). The action for the bottom row solution is S = 102 754.528, so it deviates by 0.2% from the predicted value S = 102 522.498 for this mean height given by equation (56).…”

“…In section 2 we formulate the OFM equations and boundary conditions, present a simple uniform solution of these equations, previously studied in [10,22], and argue that it describes the optimal path of the system at all λH < 0. Supercritical bifurcations of the uniform solution have recently been studied in [22]. Still, for convenience of further discussion, we briefly rederive them in section 3.…”

“…The first spatial Fourier mode to become unstable as H increases depends on the rescaled system size ℓ in a nontrivial way and is determined from equation (26). This equation was also obtained in [22] by calculating the leading-order prefactor correction to the asymptotic scaling in equation (8) through Gaussian integration of fluctuations around the uniform solution (19).…”

“…The short-time limit allows one to employ the OFM in a controlled manner [1-5, 7, 9-12, 14-20, 22], as we will reiterate shortly. The problem, defined by equations ( 1)-( 4), continues the line of study in [10,22] of finite system-size effects (which turn out to be quite dramatic) in large deviations of height of the KPZ interface. Upon rescaling t → tT, x → (νT ) 1/2 x, h → νh/λ and ξ → ( νT 3 ) −1/4 ξ, equation (1) becomes…”