Žiga Krajnik: Anomalous fluctuations in stochastic cellular automata and integrable spin chains

We introduce a discrete space-time dynamics of charged particles with stochastic particle scattering. By mapping the dynamics to a "vacany-dressed" bistochastic six-vertex model we derive the exact asymptotic anomalous (non-Gaussian) distribution of the charge current that interpolates between the charged single-file class in the limit of pure reflection and free dynamics in the limit of pure transmission. The non-Gaussianity is related to dynamical criticality by Lee-Yang analysis of the cumulant generating function. Building on macroscopic fluctuation theory, we give a hydrodynamic description of the model's anomalous fluctuations. Linear degeneracy arising from charge inertness allows for combined contributions from convective and normal diffusion.

Similar phenomenology of dynamical criticality is observed in equilibrium spin current fluctuations in the easy-axis and isotropic regimes of the XXZ spin chain. The easy-axis regime supports the non-Gaussian distribution of the charged single-file class despite not manifestly satisfying a single-file constraint. We argue anomalous fluctuations instead arise due to linear degeneracy of the vacuum polarization in the quasi-particle description. The dynamical spin structure factor at the isotropic point matches that of the Kardar-Parisi-Zhang universality class while spin fluctuations are anomalous but distinct from those of the Kardar-Parisi-Zhang universality class.