Simultaneous Random Walkers with Resetting

• Simultaneous random walkers with resetting are stochastic processes where coordinated reset events synchronize independent trajectories to produce correlated behaviors.
• The models span finite networks and continuous spaces, employing spectral techniques and memory kernels to derive steady state distributions and first-encounter statistics.
• Resetting protocols can optimize encounter times and are applicable to search problems, coordination tasks, and distributed optimization in complex environments.

Searching arXiv for papers on simultaneous random walkers with resetting and closely related resetting-induced correlation models. Simultaneous random walkers with resetting are stochastic multi-particle processes in which several walkers evolve concurrently while their reset events are coupled rather than independent. In the most direct formulation, the walkers move independently between resets, but a shared reset protocol synchronizes some aspect of their dynamics: a common reset clock, common reset times, a common reset configuration, or a common trajectory-dependent memory variable. This synchronization converts otherwise factorized dynamics into correlated many-body behavior. Recent work has treated three major realizations of this theme: diffusing particles subject to simultaneous resetting with memory in continuous space (Boyer et al., 24 Oct 2025), synchronous random walkers on finite networks analyzed through spectral methods (Suarez-Jimenez et al., 22 Apr 2026), and two-walker encounter problems on networks and in one dimension under coupled reset protocols (Rubio-Gómez et al., 7 Aug 2025, Singh et al., 2022). Across these settings, simultaneous resetting alters first-encounter statistics, generates correlations, and can induce non-equilibrium steady states or nontrivial transient regimes depending on whether the resetting is Markovian, deterministic, or memory-driven.

1. Definitions and model classes

The defining feature of simultaneous resetting is that multiple walkers share reset events. In network models, this is implemented by a global discrete-time protocol: with probability $1-\gamma$, all walkers execute one step of their individual Markov chains, while with probability $\gamma$, all walkers are synchronously reset to a prescribed configuration of home nodes (Suarez-Jimenez et al., 22 Apr 2026). For $S$ walkers on a finite graph with single-walker transition matrices $\mathbf{W}^{(s)}$, the joint no-reset evolution is the tensor product

$$\hat{\mathcal{W}}=\bigotimes_{s=1}^{S}\mathbf{W}^{(s)},$$

and simultaneous resetting replaces this by

$$\hat{\mathbf{\Pi}}(\vec r;\gamma)=(1-\gamma)\hat{\mathcal{W}}+\gamma \hat{\mathbf{\Theta}}(\vec r),$$

where $\vec r=(r_1,\dots,r_S)$ is the resetting configuration (Suarez-Jimenez et al., 22 Apr 2026).

A closely related two-walker formulation considers synchronous motion on a network, with each walker updating in parallel at every discrete time step and resetting to its own node with probability $\gamma_s$. The paper on mean first-encounter times distinguishes the case where only one walker resets from the case where both reset, but the dynamics remain simultaneous in the sense of parallel updating on the product state space (Rubio-Gómez et al., 7 Aug 2025). The encounter event is the first time both walkers occupy the same target node $j$.

In continuous space, simultaneous resetting can be trajectory-dependent rather than configuration-fixed. The model of diffusing particles evolving via simultaneous resetting with memory considers $N$ one-dimensional Brownian motions $\gamma$0, equivalently the $\gamma$1 components of an $\gamma$2-dimensional Brownian particle, all starting from the origin and all subject to a common Poisson reset clock of rate $\gamma$3 (Boyer et al., 24 Oct 2025). At a reset time $\gamma$4, a past time $\gamma$5 is drawn from a memory kernel

$\gamma$6

and every component resets simultaneously to its own position at that same past time: $\gamma$7 This is simultaneous resetting to previously visited sites rather than to a fixed point, and it interpolates between resetting to the origin and the preferential relocation or monkey-walk limit (Boyer et al., 24 Oct 2025).

A different one-dimensional realization is the lion-lamb pursuit problem in which two diffusing particles reset simultaneously to their home positions. The simultaneous reset is imposed specifically to preserve the mapping to center-of-mass and relative coordinates and to avoid spurious crossings caused by asynchronous teleportation (Singh et al., 2022).

2. Mathematical formulations

For finite networks, the simultaneous-resetting dynamics is an ordinary Markov chain on the configuration space $\gamma$8. If $\gamma$9 is the probability of being in configuration $S$0 at time $S$1 starting from $S$2, then

$S$3

(Suarez-Jimenez et al., 22 Apr 2026). This tensor-product construction is the basis for exact formulas for occupation probabilities, stationary measures, and mean first-encounter times.

The spectral structure is especially simple. If each $S$4 is diagonalizable with eigenvalues $S$5, then the no-reset joint spectrum is given by products

$S$6

Under simultaneous resetting, the leading eigenvalue remains $S$7, while every non-leading eigenvalue is multiplied by $S$8: $S$9 This gives a direct spectral interpretation of how synchronous resetting damps non-stationary modes (Suarez-Jimenez et al., 22 Apr 2026).

For the two-walker encounter problem on networks, the same logic applies at the level of $\mathbf{W}^{(s)}$0, or of $\mathbf{W}^{(s)}$1 when resetting is present (Rubio-Gómez et al., 7 Aug 2025). The mean first-encounter time at node $\mathbf{W}^{(s)}$2 is then written as an explicit double spectral sum over the eigenvalues and biorthogonal eigenvectors of the single-walker transition matrix.

In the memory-driven Brownian model, the dynamics is non-Markovian in time but still admits a closed Fokker–Planck description for the joint density $\mathbf{W}^{(s)}$3: $\mathbf{W}^{(s)}$4 (Boyer et al., 24 Oct 2025). The Fourier transform obeys an evolution equation whose special cases include the Markovian reset-to-origin limit and the preferential-relocation limit. In the latter case, the exact Fourier-space solution involves Kummer’s confluent hypergeometric function (Boyer et al., 24 Oct 2025).

The lion-lamb model uses a different reduction. With $\mathbf{W}^{(s)}$5 and $\mathbf{W}^{(s)}$6, the center of mass is a free Brownian motion and the relative coordinate is a Brownian motion with absorption at $\mathbf{W}^{(s)}$7. Simultaneous resetting to the home positions sends $\mathbf{W}^{(s)}$8 and $\mathbf{W}^{(s)}$9, converting the encounter problem into a first-passage problem with reset in the relative coordinate and a reset diffusion in the center-of-mass coordinate (Singh et al., 2022).

3. Correlation generation and conditional structure

A central result of the memory-driven continuous-space model is that simultaneous resetting generates correlations even when the underlying motions are independent Brownian processes (Boyer et al., 24 Oct 2025). By symmetry, $$\hat{\mathcal{W}}=\bigotimes_{s=1}^{S}\mathbf{W}^{(s)},$$0, so the relevant correlation observable is built from squared displacements: $$\hat{\mathcal{W}}=\bigotimes_{s=1}^{S}\mathbf{W}^{(s)},$$1 which lies in $$\hat{\mathcal{W}}=\bigotimes_{s=1}^{S}\mathbf{W}^{(s)},$$2 (Boyer et al., 24 Oct 2025). The mechanism is that all walkers share the same reset times and, in the memory-dependent model, the same sampled past time $$\hat{\mathcal{W}}=\bigotimes_{s=1}^{S}\mathbf{W}^{(s)},$$3. Conditioned on the effective Brownian duration $$\hat{\mathcal{W}}=\bigotimes_{s=1}^{S}\mathbf{W}^{(s)},$$4, the components are independent and identically distributed Gaussians, but averaging over the random $$\hat{\mathcal{W}}=\bigotimes_{s=1}^{S}\mathbf{W}^{(s)},$$5 produces nontrivial correlations in amplitudes (Boyer et al., 24 Oct 2025).

This conditional i.i.d. structure has an exact representation: $$\hat{\mathcal{W}}=\bigotimes_{s=1}^{S}\mathbf{W}^{(s)},$$6 where $$\hat{\mathcal{W}}=\bigotimes_{s=1}^{S}\mathbf{W}^{(s)},$$7 is the free Brownian propagator and $$\hat{\mathcal{W}}=\bigotimes_{s=1}^{S}\mathbf{W}^{(s)},$$8 is the distribution of an effective Brownian path duration (Boyer et al., 24 Oct 2025). In the Markovian reset-to-origin case, $$\hat{\mathcal{W}}=\bigotimes_{s=1}^{S}\mathbf{W}^{(s)},$$9 is the time since the last reset; in the non-Markovian memory case, $$\hat{\mathbf{\Pi}}(\vec r;\gamma)=(1-\gamma)\hat{\mathcal{W}}+\gamma \hat{\mathbf{\Theta}}(\vec r),$$0 is the duration of a continuous Brownian path built by gluing together trajectory segments connected by resets to previously visited points (Boyer et al., 24 Oct 2025). This representation unifies the Markovian and memory-driven regimes and makes clear that simultaneous resetting acts as a common random environment.

A related but distinct source of correlation appears in the density-dependent resetting framework for many walkers on networks. There the walkers are non-interacting during transport, but the resetting fraction at node $$\hat{\mathbf{\Pi}}(\vec r;\gamma)=(1-\gamma)\hat{\mathcal{W}}+\gamma \hat{\mathbf{\Theta}}(\vec r),$$1 is

$$\hat{\mathbf{\Pi}}(\vec r;\gamma)=(1-\gamma)\hat{\mathcal{W}}+\gamma \hat{\mathbf{\Theta}}(\vec r),$$2

so the reset probability of each walker depends on local density and therefore on the positions of the others (Coghi et al., 2024). The paper explicitly states that density-dependent resetting induces correlations in otherwise non-interacting walkers. This is not identical to synchronous reset clocks, but it belongs to the broader class of collective resetting rules in which common reset mechanisms produce effective interactions (Coghi et al., 2024).

4. Encounter times, capture statistics, and optimal resetting

A major line of work studies simultaneous resetting through first-encounter observables. On finite networks, the mean first-encounter time (MFET) is the expected time required for all walkers to meet for the first time at a given node (Suarez-Jimenez et al., 22 Apr 2026). For two or more walkers, exact formulas are obtained in terms of the eigenvalues and eigenvectors of the no-reset transition matrices (Suarez-Jimenez et al., 22 Apr 2026). The dependence on the reset probability $$\hat{\mathbf{\Pi}}(\vec r;\gamma)=(1-\gamma)\hat{\mathcal{W}}+\gamma \hat{\mathbf{\Theta}}(\vec r),$$3 enters only through denominators of the form $$\hat{\mathbf{\Pi}}(\vec r;\gamma)=(1-\gamma)\hat{\mathcal{W}}+\gamma \hat{\mathbf{\Theta}}(\vec r),$$4 and through the stationary distribution under resetting. This makes it possible to derive a general criterion for when resetting is beneficial.

For resetting to the initial configuration, beneficial resetting is characterized by the no-reset coefficient of variation. If $$\hat{\mathbf{\Pi}}(\vec r;\gamma)=(1-\gamma)\hat{\mathcal{W}}+\gamma \hat{\mathbf{\Theta}}(\vec r),$$5 is the first-encounter time without reset, then an infinitesimal amount of simultaneous resetting reduces the MFET precisely when

$$\hat{\mathbf{\Pi}}(\vec r;\gamma)=(1-\gamma)\hat{\mathcal{W}}+\gamma \hat{\mathbf{\Theta}}(\vec r),$$6

equivalently when the quantity

$$\hat{\mathbf{\Pi}}(\vec r;\gamma)=(1-\gamma)\hat{\mathcal{W}}+\gamma \hat{\mathbf{\Theta}}(\vec r),$$7

is positive (Suarez-Jimenez et al., 22 Apr 2026). This criterion depends only on the no-reset process. At the optimal reset probability $$\hat{\mathbf{\Pi}}(\vec r;\gamma)=(1-\gamma)\hat{\mathcal{W}}+\gamma \hat{\mathbf{\Theta}}(\vec r),$$8, the first two moments satisfy

$$\hat{\mathbf{\Pi}}(\vec r;\gamma)=(1-\gamma)\hat{\mathcal{W}}+\gamma \hat{\mathbf{\Theta}}(\vec r),$$9

(Suarez-Jimenez et al., 22 Apr 2026).

The explicit two-walker network formulas show the same non-monotonic phenomenon. When one walker resets and the other does not, or when both reset, the MFET at a target node $\vec r=(r_1,\dots,r_S)$0 is an analytic function of the reset rates and the network spectrum (Rubio-Gómez et al., 7 Aug 2025). Numerical studies on rings, Cayley trees, and random networks generated using the Erdős–Rényi, Watts–Strogatz, and Barabási–Albert algorithms show that resetting can significantly reduce encounter times, but only for some targets and reset locations (Rubio-Gómez et al., 7 Aug 2025). A plausible implication is that resetting is most helpful when the reset nodes lie in spectrally central regions relative to the target.

The lion-lamb model exhibits the same logic in continuous space. Without resetting, the capture location distribution is a Cauchy law centered at $\vec r=(r_1,\dots,r_S)$1 (Singh et al., 2022). Under Poissonian simultaneous resetting to the home positions, the capture location distribution decomposes into a Cauchy-like central part from no-reset trajectories and exponential tails from trajectories with at least one reset (Singh et al., 2022). Under sharp deterministic resetting, the central contribution becomes effectively Gaussian while the tails remain exponential (Singh et al., 2022). The mean capture time becomes finite in both reset protocols, with explicit formulas. For the parameter choice $\vec r=(r_1,\dots,r_S)$2, $\vec r=(r_1,\dots,r_S)$3, the optimal Poissonian resetting rate is $\vec r=(r_1,\dots,r_S)$4 with minimal mean capture time $\vec r=(r_1,\dots,r_S)$5, whereas the optimal sharp reset period is $\vec r=(r_1,\dots,r_S)$6 with minimal mean capture time $\vec r=(r_1,\dots,r_S)$7, so sharp resetting is more efficient in that example (Singh et al., 2022).

A different first-passage perspective comes from simultaneous resetting of $\vec r=(r_1,\dots,r_S)$8 Brownian searchers to a common initial position. When any walker first hits the origin, the search ends. The mean first-passage time has a minimum at a nonzero reset rate only below a critical number of walkers; above that, resetting is detrimental (Biroli et al., 2023). For independent resetting the critical value is

$\vec r=(r_1,\dots,r_S)$9

while for simultaneous resetting it is

$\gamma_s$0

(Biroli et al., 2023). Thus simultaneous resetting becomes harmful at a smaller walker number than independent resetting, which the paper attributes to the stronger dynamical correlations created by global resets (Biroli et al., 2023).

5. Memory, non-Markovian resetting, and asymptotic regimes

The memory parameter $\gamma_s$1 in the simultaneous Brownian model controls the range of the resetting memory kernel and interpolates between two limits (Boyer et al., 24 Oct 2025). In the Markovian limit $\gamma_s$2, resetting always returns the walkers to the origin; in the preferential-relocation limit $\gamma_s$3, the reset time is sampled uniformly over the entire past. The correlation coefficient $\gamma_s$4 behaves qualitatively differently in these limits.

For pure resetting to the origin, the correlation coefficient increases monotonically from $\gamma_s$5 and converges exponentially to

$\gamma_s$6

which is identified as a universal bound for the non-equilibrium steady state of diffusion with simultaneous resetting to a single point (Boyer et al., 24 Oct 2025). For preferential relocation, $\gamma_s$7 is non-monotonic: it grows from zero, reaches a maximum at

$\gamma_s$8

and then decays to zero as

$\gamma_s$9

with $j$0 (Boyer et al., 24 Oct 2025). The decay is logarithmically slow, so the asymptotic decorrelation is extremely gradual.

For general finite $j$1, there is a transition between monotonic and non-monotonic correlation dynamics at the critical dimensionless memory strength

$j$2

(Boyer et al., 24 Oct 2025). If $j$3, the correlation coefficient has a finite-time maximum; if $j$4, it increases monotonically to its asymptotic value. For every $j$5, the process reaches a non-equilibrium steady state with finite asymptotic correlation $j$6, and this asymptotic value obeys

$j$7

(Boyer et al., 24 Oct 2025).

The preferential-relocation limit also displays anomalous spreading. The mean-square displacement grows only as

$j$8

there is no non-equilibrium steady state, and the joint PDF remains time-dependent but asymptotically Gaussian in a scaling form (Boyer et al., 24 Oct 2025). This strongly contrasts with the exponentially localized stationary states familiar from Markovian resetting to fixed points.

Non-Markovian resetting on networks has analogous distinctions. In the single-walker network framework with general renewal resetting, light-tailed inter-reset distributions produce a non-equilibrium steady state, whereas fat-tailed cases such as Sibuya resetting with infinite mean inter-reset time do not (Michelitsch et al., 2024). This suggests a broader principle: whether simultaneous resetting creates stationary many-body structure depends not only on the spatial reset rule but also on the renewal statistics of reset times.

6. Topology, synchronization, and comparative performance

The impact of simultaneous resetting depends strongly on topology. On rings, regular and homogeneous structure favors synchronized resetting: for two walkers starting from $j$9, $N$0 on a ring of size $N$1, simultaneous resetting never performs worse than independent resetting in terms of optimal MFET, although for some target nodes both protocols are no better than no resetting (Suarez-Jimenez et al., 22 Apr 2026). On heterogeneous networks, especially Barabási–Albert graphs with hubs, some nodes exhibit the opposite ordering and independent resetting can outperform simultaneous resetting (Suarez-Jimenez et al., 22 Apr 2026).

This contrast is quantified by the difference

$N$2

Positive values indicate an advantage for simultaneous resetting; negative values indicate an advantage for independent resetting (Suarez-Jimenez et al., 22 Apr 2026). The interpretation offered is a trade-off between synchronization and exploration diversity. Homogeneous graphs favor synchronization because coordinated returns keep trajectories aligned toward a common node, while heterogeneous graphs can reward exploration diversity because independent resets allow walkers to sample structurally different regions (Suarez-Jimenez et al., 22 Apr 2026).

The same topological dependence appears in the more specific two-walker encounter model on networks. Resetting significantly reduces encounter times on rings, Cayley trees, Erdős–Rényi networks, Watts–Strogatz networks, and Barabási–Albert networks, but the optimal rates and the degree of improvement depend on the target node and reset configuration (Rubio-Gómez et al., 7 Aug 2025). On Barabási–Albert graphs, resetting to hubs can be especially effective because the underlying random walk visits hubs frequently (Rubio-Gómez et al., 7 Aug 2025).

A broader comparison with independent protocols is also available in diffusive search on the line. The critical number of walkers above which resetting hinders search is smaller for simultaneous resetting than for independent resetting, again indicating that synchrony reduces trajectory diversity (Biroli et al., 2023). This should not be misread as implying that simultaneous resetting is generically inferior. Rather, the evidence indicates that its value is context-dependent: it can be optimal for encounter tasks on homogeneous substrates or nearby targets, yet suboptimal for highly exploratory search with many walkers or in strongly heterogeneous environments (Suarez-Jimenez et al., 22 Apr 2026, Biroli et al., 2023).

7. Generalizations, applications, and open directions

The study of simultaneous random walkers with resetting now spans several adjacent domains. On finite networks, the spectral formalism accommodates local and nonlocal walks, including mixtures of nearest-neighbor random walks and Lévy-flight–type transitions defined via fractional Laplacians (Suarez-Jimenez et al., 22 Apr 2026). For a ring of size $N$3, combining one normal walker and one Lévy walker shows that near targets favor more local dynamics and moderate resetting, while distant targets can favor strongly nonlocal dynamics together with small resetting (Suarez-Jimenez et al., 22 Apr 2026). This establishes that simultaneous resetting and transport nonlocality must be optimized jointly.

In continuous space, the memory-driven Brownian model connects simultaneous resetting to preferential relocation and to animal movement models. The two-dimensional case is suggested as a model of animals revisiting familiar locations, with the correlation structure between spatial components potentially serving as a diagnostic of memory use and spatial reuse (Boyer et al., 24 Oct 2025). The same paper notes that the conditional i.i.d. representation may facilitate exact studies of extreme-value statistics and spacing distributions in strongly correlated resetting systems (Boyer et al., 24 Oct 2025).

Applications mentioned across the literature include rendezvous problems, coordinated search and rescue, distributed optimization, epidemic spreading, and human mobility (Suarez-Jimenez et al., 22 Apr 2026, Rubio-Gómez et al., 7 Aug 2025). The key technical reason is that simultaneous resetting naturally couples arrival times and spatial configurations without introducing direct inter-particle forces, making it a tractable mechanism for controlled coordination.

Several open directions recur across the papers. One is the extension from Markovian to non-Markovian reset protocols on networks (Suarez-Jimenez et al., 22 Apr 2026). Another is the incorporation of interactions beyond shared resets, such as exclusion or attraction, on top of synchronous resetting with memory (Boyer et al., 24 Oct 2025). Higher-dimensional and field-like versions, where entire interfaces or spatial fields reset simultaneously, are also explicitly suggested (Boyer et al., 24 Oct 2025). A plausible implication is that the conditional-i.i.d. perspective and the tensor-product spectral framework could serve as the two main organizing principles for such extensions: the former for continuous-space memory models, the latter for finite-state synchronous Markov processes.

At a conceptual level, simultaneous random walkers with resetting provide a bridge between independent random motion and genuinely collective stochastic dynamics. The walkers may remain independent between resets, yet a shared reset rule can generate measurable correlations, reshape first-passage laws, create or destroy steady states, and alter optimal search strategies. The recent literature shows that these effects are not peripheral corrections but organizing features of the dynamics itself (Boyer et al., 24 Oct 2025, Suarez-Jimenez et al., 22 Apr 2026, Rubio-Gómez et al., 7 Aug 2025).