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arxiv: 2505.19903 · v4 · submitted 2025-05-26 · ❄️ cond-mat.stat-mech · physics.data-an

Diffusion with stochastic resetting on a lattice

Alexander K. Hartmann , Satya N. Majumdar This is my paper

Pith reviewed 2026-05-19 13:42 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.data-an
keywords stochastic resettingmean first-passage timelattice diffusionhypercubic latticenonequilibrium stationary stateoptimal resetting rate
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The pith

An exact formula for the mean first-passage time is derived for a diffusing particle on a lattice that resets to its starting position at rate r.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an exact closed-form expression for the mean time a random walker on a d-dimensional hypercubic lattice takes to reach a fixed target when it resets back to its initial site at Poissonian rate r. This expression holds for arbitrary resetting rate and arbitrary lattice starting position, allowing exploration of regimes where continuum approximations break down. It shows that the mean time diverges both as r approaches zero and as r approaches infinity when the start is not adjacent to the target, with a minimum at an intermediate rate, while nearest-neighbor starts instead favor ever-higher resetting rates. The work also gives the exact nonequilibrium stationary position distribution in the absence of a target. A reader would care because the results describe search processes on discrete structures that arise in many physical and biological settings.

Core claim

We provide an exact formula for the mean first-passage time (MFPT) to a target at the origin for a single particle diffusing on a d-dimensional hypercubic lattice starting from a fixed initial position vec R_0 and resetting to vec R_0 with a rate r. For starting points that are not nearest neighbors, the MFPT diverges as r^phi with phi equal to the sum of absolute coordinate indices minus one when r goes to infinity. When the starting point is a nearest neighbor, the MFPT decreases monotonically to the universal value 1 as r goes to infinity. The continuum limit is recovered when r to zero and R_0 to infinity are taken with sqrt(r) R_0 held fixed. In the absence of a target the exact nonequl

What carries the argument

The exact MFPT obtained by solving translationally invariant recurrence relations on the infinite lattice via generating functions, which are made possible by the combination of unbiased nearest-neighbor jumps and Poissonian resets to a fixed origin.

If this is right

  • For non-nearest-neighbor starting positions the MFPT as a function of r has a unique minimum at some finite resetting rate.
  • As r tends to infinity the MFPT grows as a power law whose exponent equals the Manhattan distance from the target minus one.
  • When the walker starts adjacent to the target the MFPT falls monotonically toward the universal limit of one.
  • The continuum-space results are recovered in the joint limit of small r and large R_0 with sqrt(r) R_0 fixed.
  • Without a target the stationary position distribution exhibits lattice-specific features absent from the continuum description.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Manhattan-distance dependence of the high-rate exponent suggests that optimal resetting protocols on grids are governed by discrete geometry rather than Euclidean distance.
  • The universal limiting value of one for nearest-neighbor starts may generalize to other discrete Markovian resetting models.
  • Verification up to fifty dimensions indicates the lattice formulas remain useful in regimes where continuum theory is inaccurate.
  • The exact stationary distribution offers a benchmark for testing approximate theories of resetting in confined or disordered lattices.

Load-bearing premise

The lattice is infinite with no boundaries and the walker performs unbiased nearest-neighbor jumps interrupted by Poissonian resets to the fixed initial position.

What would settle it

Numerical measurement of the MFPT at large resetting rate r for a starting lattice vector with coordinate sum S, checking whether the time grows proportionally to r raised to the power S minus one.

Figures

Figures reproduced from arXiv: 2505.19903 by Alexander K. Hartmann, Satya N. Majumdar.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The simulation results [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The number [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Markov chains for representing lattices in [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Non-equilibrium steady-state density for dimension [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

We provide an exact formula for the mean first-passage time (MFPT) to a target at the origin for a single particle diffusing on a $d$-dimensional hypercubic {\em lattice} starting from a fixed initial position $\vec R_0$ and resetting to $\vec R_0$ with a rate $r$. Previously known results in the continuous space are recovered in the scaling limit $r\to 0$, $R_0=|\vec R_0|\to \infty$ with the product $\sqrt{r}\, R_0$ fixed. However, our formula is valid for any $r$ and any $\vec R_0$ that enables us to explore a much wider region of the parameter space that is inaccessible in the continuum limit. For example, we have shown that the MFPT, as a function of $r$ for fixed $\vec R_0$, diverges in the two opposite limits $r\to 0$ and $r\to \infty$ with a unique minimum in between, provided the starting point is not a nearest neighbour of the target. In this case, the MFPT diverges as a power law $\sim r^{\phi}$ as $r\to \infty$, but very interestingly with an exponent $\phi= (|m_1|+|m_2|+\ldots +|m_d|)-1$ that depends on the starting point $\vec R_0= a\, (m_1,m_2,\ldots, m_d)$ where $a$ is the lattice spacing and $m_i$'s are integers. If, on the other hand, the starting point happens to be a nearest neighbour of the target, then the MFPT decreases monotonically with increasing $r$, approaching a universal limiting value $1$ as $r\to \infty$, indicating that the optimal resetting rate in this case is infinity. We provide a simple physical reason and a simple Markov-chain explanation behind this somewhat unexpected universal result. Our analytical predictions are verified in numerical simulations on lattices up to $50$ dimensions. Finally, in the absence of a target, we also compute exactly the position distribution of the walker in the nonequlibrium stationary state that also displays interesting lattice effects not captured by the continuum theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript derives an exact closed-form expression for the mean first-passage time (MFPT) to the origin for a single unbiased nearest-neighbor random walker on an infinite d-dimensional hypercubic lattice that resets to its fixed initial position R0 at Poisson rate r. The formula is obtained from the standard generating-function or recurrence approach for the first-passage problem on the lattice; it recovers the known continuum MFPT in the joint limit r→0, |R0|→∞ with √r |R0| fixed. Lattice-specific results include a power-law divergence MFPT ∼ r^φ with φ = ∑|mi|−1 (Manhattan distance minus one) for large r when the starting site is not a nearest neighbor, a monotonic decrease to the universal value 1 when the start is a nearest neighbor, and an exact expression for the nonequilibrium stationary-state occupation probabilities in the absence of a target. Predictions are checked against direct Monte Carlo simulations on lattices up to d=50.

Significance. If the claimed exact formula is correct, the work supplies a concrete, parameter-free extension of resetting diffusion to discrete lattices that reveals qualitative features (Manhattan-distance exponent, nearest-neighbor universality) invisible in the continuum limit. The high-dimensional numerical verification and the exact stationary-state distribution constitute reproducible, falsifiable content that strengthens the central claim.

major comments (1)
  1. [§3] §3 (MFPT derivation): the recurrence relation for the MFPT is stated to close after a single generating-function step, but the explicit inversion that yields the closed expression for arbitrary R0 = a(m1,…,md) is not shown; without this step it is unclear whether the final formula remains free of unevaluated lattice sums or requires truncation for d>2.
minor comments (3)
  1. [Abstract] Abstract: 'nonequlibrium' is a typographical error for 'nonequilibrium'.
  2. [Figure 2] Figure 2 caption: the legend does not indicate the lattice spacing a or the precise definition of the Manhattan distance used for the φ exponent.
  3. [§4] §4 (stationary state): the normalization of the occupation probabilities is stated but the explicit sum over all lattice sites that confirms ∑P(r) = 1 is omitted; adding this short verification would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive assessment of its significance, and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (MFPT derivation): the recurrence relation for the MFPT is stated to close after a single generating-function step, but the explicit inversion that yields the closed expression for arbitrary R0 = a(m1,…,md) is not shown; without this step it is unclear whether the final formula remains free of unevaluated lattice sums or requires truncation for d>2.

    Authors: We appreciate the referee drawing attention to the presentation of the derivation. The recurrence is solved via generating functions, and the closed-form MFPT for general R0 = a(m1, …, md) follows from a standard inversion that expresses the result in terms of the d-dimensional lattice Green function G_d(z; m1, …, md) evaluated at z determined by the resetting rate r. This inversion yields an explicit expression involving only a finite combination of Green-function values at the coordinate shifts fixed by the integers mi; no unevaluated infinite sums remain and no truncation is introduced for any d, including d > 2. The same representation is used in the high-dimensional Monte Carlo checks up to d = 50. In the revised manuscript we will insert the explicit inversion algebra immediately after the generating-function equation to make this step transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the MFPT using standard translationally invariant recurrence relations and generating-function methods for nearest-neighbor walks on infinite hypercubic lattices with Poissonian resetting. These techniques are independent, well-established tools that do not presuppose the target result. The derivation explicitly recovers the known continuum limit under the stated scaling, yields lattice-specific features (such as the Manhattan-distance exponent φ) that follow directly from the same equations, and is cross-checked by numerical simulations up to d=50. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the central claim. The model assumptions are stated upfront and the solutions are obtained without circular reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim of an exact MFPT formula rests on solving the appropriate master equation or using combinatorial methods for first passage times with resetting on an infinite lattice, using standard mathematical techniques without additional free parameters.

axioms (1)
  • domain assumption The diffusion is modeled as a continuous-time random walk on the hypercubic lattice with nearest-neighbor jumps between resets.
    This is the standard setup for lattice diffusion mentioned in the abstract.

pith-pipeline@v0.9.0 · 5956 in / 1317 out tokens · 142748 ms · 2026-05-19T13:42:32.164806+00:00 · methodology

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Reference graph

Works this paper leans on

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