Emerging correlations between diffusing particles evolving via simultaneous resetting with memory

TL;DR

This paper addresses how diffusive components in N dimensions become correlated when resetting occurs to past visited positions with memory. It develops a rigorous, memory-aware framework to quantify correlations via a time-dependent a(t) derived from the small-k expansion of the joint PDF, and solves the diffusion problem for an exponential memory kernel φ(τ)=e^{−λτ}, covering λ→∞ (reset to origin), λ→0 (preferential relocation), and finite λ. Key findings show a monotone buildup to a fixed a(∞)=1/5 in the memoryless-like limit, a nonmonotonic finite-time peak with a0(z^*)≈0.0996 in the preferential relocation case, and a λ-dependent transition in the general case with a critical memory Λ_c≈0.0743 marking a change from nonmonotonic to monotonic behavior; the work further reveals a unifying renewal-like, c.i.i.d. structure that underpins the emergence of correlations across memory regimes. The results advance understanding of how memory and simultaneous resetting shape correlations in high-dimensional diffusion, with implications for interpreting memory use in ecological and physical systems and for designing experiments to probe non-Markovian resetting mechanisms.

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 $t$.

Figures (4)

• Figure 1: (a) Correlation coefficient $a_{\lambda}(z)$ defined in Eq. (\ref{['ac2c4']}) as a function of the rescaled time $z=rt$, for various particular values of $\lambda$. The case $\lambda=\infty$ (resetting to the origin) is given by Eq. (\ref{['alambdainfty']}) and tends exponentially fast to the asymptotic value $1/5$. In the case $\lambda=0$ (preferential relocation model), $a_0(z)$ is obtained from Eqs. (\ref{['var.1b']})-(\ref{['num.1b']}) and exhibits a non-monotonous behaviour with a maximum at $z^*=14.6732\ldots$. The correlation $a_0(z)$ increases linearly at small $z$ and, as shown by Panel (b) on a different scale of values of $z$, decays very slowly to $0$ at large $z$, as $1/\ln z$ to leading order. For all $\lambda>0$, $a_{\lambda}(z)$ tends to finite value as $z\to\infty$, which depends on $\lambda/r$ only and is given by Eqs. (\ref{['azinfty']})-(\ref{['fy']}). For $\lambda/r <\Lambda_c=0.0743\ldots$, $a_{\lambda}(z)$ has a maximum at a finite $z^*(\lambda/r)$, whereas for $\lambda/r\ge\Lambda_c$, $a_{\lambda}(z)$ increases monotonously with $z$ like in the case $\lambda=\infty$.
• Figure 2: Asymptotic value of the correlation at large times as a function of $\Lambda=\lambda/r$.
• Figure 3: Rescaled time $z^*$ where the correlation $a_{\lambda}(z)$ is maximal, depicted in Fig. \ref{['az.fig']}, as a function of $\lambda/r$ (in log scale). For $\lambda/r<\Lambda_c=0.0743\ldots$, $z^*$ is finite and $a_{\lambda}(z)$ non-monotonous. For $\lambda/r>\Lambda_c$, or a memory kernel sufficiently peaked at short times, $a_{\lambda}(z)$ increases monotonously with $z$ and the maximal correlation is reached at infinite times ($z^*=\infty$). The vertical dashed line shows the location of the critical value $\ln(\Lambda_c)=\ln(0.0743\cdots)=-2.59964\ldots$.
• Figure 4: Illustration of a two-dimensional diffusion process after 3 resetting events to previous positions, or 4 consecutive paths. The long-range jumps to previous points of the trajectory are shown by dashed arrows. The position $\vec{x}(t)$ of the particle at time $t$ can also be reached by following the continuous Brownian path ${\cal P}$ (in red), which is of much shorter duration $t'$. The components of the position $\vec{x}(t)$ have a c.i.i.d. structure.