Scaling limits for Rudvalis card shuffles

TL;DR

This paper investigates the Rudvalis card shuffle and its variations, deriving hydrodynamic limits and studying fluctuations by projecting them onto stochastic particle systems, revealing different behaviors under symmetric and asymmetric conditions.

Contribution

It introduces a projection of Rudvalis shuffles onto particle systems and derives their hydrodynamic limits and fluctuation properties.

Findings

Asymmetric Rudvalis shuffle leads to a transport equation in hydrodynamic limit.

Symmetric and weakly asymmetric shuffles exhibit diffusive behavior described by a martingale problem.

Hydrodynamic limits depend on the symmetry and exchange rates of the system.

Abstract

We consider the Rudvalis card shuffle and some of its variations that were introduced by Diaconis and Saloff-Coste in \cite{symmetrized}, and we project them to some stochastic interacting particle system. For the latter, we derive the hydrodynamic limits and we study the equilibrium fluctuations. Our results show that, for these shuffles, when we consider an asymmetric variation of the Rudvalis shuffle, the hydrodynamic limit in Eulerian scale is given in terms of a transport equation with a constant that depends on the exchange rates of the system; while for symmetric and weakly asymmetric variations of this shuffle, in diffusive time scale, the evolution is given by the solution of a martingale problem.