Emerging correlations between diffusing particles evolving via simultaneous resetting with memory

Abstract

We study the emergence of correlations between $N$ components of the position of a diffusive walker in $N$ dimensions that starts at the origin and resets to previously visited sites with certain probabilities. This is equivalent to $N$ independent one-dimensional diffusive processes starting from the origin and being subject to simultaneous resetting to positions visited in the past. Resetting follows a memory kernel that interpolates between resetting to the origin only, and the preferential relocation model, a path-dependent process which is highly non-Markov. For weak memory, the correlation coefficient between two components of the $N$-dimensional process grows monotonously with time and tends at late times to a constant bounded by $1/5$, the value corresponding to the non-equilibrium steady state of resetting to the origin. When memory is sufficiently long-ranged, the correlation is non-monotonous and reaches a maximum at a finite time before converging to its asymptotic value. These two regimes are separated by a critical memory parameter value. In the limiting case of the preferential relocation model, the components become uncorrelated at both short and long times, but the correlation vanishes logarithmically slowly at late times. The emergence of correlations through resetting can be described in a unified way in all cases by noticing that the processes are conditionally independent and identically distributed, even in the presence of memory. In the non-Markovian case, the conditioning parameter is the duration of a Brownian path composed of several parts of the full trajectory of a fixed duration $t$.