Nonisospectral Integrability and Exact Current Fluctuations in the Two-Dimensional SSEP

Abstract

We study annealed current fluctuations in the two-dimensional symmetric simple exclusion process (SSEP) across a circular passive counting boundary. The initial average density is $\rho_1$ inside a disk of radius $R$ and $\rho_2$ outside, and the observable is the net decrease of the particle number in the disk over a finite time. Using convexity of the macroscopic fluctuation theory action and rotational averaging, we show that the minimizer of the full two-dimensional variational problem may be chosen radially symmetric. For the resulting radial problem, a suitable change of variables leads to a nonisospectral formulation on the half-line. Combining the associated scattering construction with a scalar factorization, we obtain a closed expression for the scaled cumulant generating function and hence the annealed large-deviation statistics of the current.