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arxiv: 2512.24359 · v2 · submitted 2025-12-30 · ❄️ cond-mat.stat-mech

Tethering effects on first-passage variables of lattice random walks in linear and quadratic focal point potentials

Debraj Das , Luca Giuggioli This is my paper

Pith reviewed 2026-05-16 18:54 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords lattice random walkfirst-passage timeV-shaped potentialU-shaped potentialresettingoccupation probabilitydistinct sites visited
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0 comments X

The pith

Lattice random walks under linear focal potentials grow the number of distinct sites visited only logarithmically and can show a minimum mean first-passage time at intermediate bias.

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

The paper studies lattice random walks pulled toward a focal point by discretised linear (V-shaped) or quadratic (U-shaped) potentials. It derives the generating function for the occupation probability in the unbounded linear case and uses it to show that the mean number of distinct sites visited increases only logarithmically with time. First-passage probabilities are analysed for both potentials, revealing that their means can reach a minimum when bias strength is varied, with the location depending on where the walker starts and where the target lies relative to the focal point. Adding stochastic resetting to a non-focal site in bounded versions produces different steady-state distributions for the two potentials and induces a motion-limited regime in first-passage dynamics even at moderate resetting rates.

Core claim

For the unbounded V-potential the generating function of the occupation probability is obtained, from which the mean number of distinct sites visited is shown to grow logarithmically at long times. The mean first-passage time to a target site displays a minimum as a function of bias strength when the initial and target sites lie on opposite sides of the focal point or at certain distances from it. Qualitatively similar non-monotonic dependence appears for the finite U-potential. When resetting is superimposed on bounded V- and U-potentials, the steady-state probabilities and first-passage dynamics differ between the two shapes, with a motion-limited regime emerging for moderate resetting.

What carries the argument

The position-dependent bias obtained by discretising linear and quadratic focal potentials on the lattice, together with generating-function techniques for occupation probabilities and first-passage statistics.

If this is right

  • The mean number of distinct sites visited grows only logarithmically rather than linearly with time in the unbounded linear potential.
  • Mean first-passage times can be minimised by tuning bias strength for given initial and target locations relative to the focal point.
  • Resetting produces qualitatively different steady-state occupation for bounded linear versus quadratic potentials.
  • A motion-limited regime in first-passage dynamics appears even for moderate resetting probabilities.

Where Pith is reading between the lines

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

  • The logarithmic growth implies that linear tethers restrict spatial exploration far more severely than unbiased diffusion or quadratic confinement.
  • The existence of an optimal bias for minimal mean first-passage time suggests a trade-off between drift speed and diffusive spread that could be tested by varying lattice spacing.
  • Combining resetting with focal potentials offers a minimal model for biological search processes that return to a home base after excursions.

Load-bearing premise

Discretising the continuous linear and quadratic potentials onto the lattice preserves the qualitative bias structure without changing long-time asymptotics or the existence of a minimum in mean first-passage time.

What would settle it

Numerical simulations on large lattices that show the mean number of distinct sites growing faster than logarithmically, or that find no minimum in mean first-passage time for any bias strength and any choice of initial and target sites, would contradict the central claims.

Figures

Figures reproduced from arXiv: 2512.24359 by Debraj Das, Luca Giuggioli.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic diagram of transition probabilities for a [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Propagator [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. First-passage walker statistics in a V–potential cen [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Average number of distinct sites visited [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Schematic diagram of transition probabilities for a [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Propagator [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison between the mean first-passage time [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Steady-state probability distributions in the presence [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. First-passage probability starting from the resetting [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
read the original abstract

Diffusion in a confining potential offers a minimal setting to understand the interplay between random motion and deterministic forces driving a particle towards a focal point or potential minimum. In continuous space and time, two extensively studied examples are Brownian motion in a linear (V-shaped) or a quadratic (U-shaped) potential. The deterministic bias towards the minimum is represented, respectively, by a constant force for the former and by an elastic restoring force that increases proportionally with distance for the latter. Surprisingly, unlike Brownian walks, random walks under focal point potentials in discrete space and time have received little attention. Here, we bridge this gap by analysing the dynamics of lattice random walkers in the presence of a V-shaped potential, both in a finite and an infinite spatial domain, and a finite U-shaped potential. For the V-potential in unbounded space, we find the generating function of the occupation probability and analyse the time dependence of the mean number of distinct sites visited, demonstrating that its long-time growth is logarithmic. We also study the first-passage probability and show that its mean may display a minimum as a function of bias strength, depending on the location of the initial and target sites relative to the focal point. Qualitatively similar dependencies in the first-passage probability and its mean appear for the finite U-potential. As a comparative analysis to the U-potential, we construct the bounded V-potential and superimpose in both cases a resetting process, in which the walker returns at random times to a site distinct from the focal point with some probability. We quantify the different effects of resetting on the steady-state probability and the first-passage dynamics in the two cases, and show a motion-limited regime emerges even for relatively moderate resetting probabilities.

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

2 major / 3 minor

Summary. The manuscript examines lattice random walks in linear (V-shaped) and quadratic (U-shaped) focal-point potentials. For the unbounded V-potential it derives the generating function of the occupation probability and shows that the mean number of distinct sites visited grows logarithmically at long times; it further analyzes the first-passage probability and its mean, which can exhibit a minimum versus bias strength depending on initial and target locations. Comparative results are given for the finite U-potential and for both potentials under resetting, quantifying how resetting alters the steady-state distribution and first-passage dynamics.

Significance. If the derivations are correct, the work supplies exact, checkable results for the discrete-space versions of two canonical confined-diffusion problems. The logarithmic range growth, arising from the geometric stationary measure, and the non-monotonic mean-first-passage behavior constitute concrete, falsifiable predictions that distinguish the lattice setting from its continuous counterpart. The resetting analysis further illustrates how an external resetting mechanism interacts with the potential shape.

major comments (2)
  1. [§3] The central derivation of the generating function for the unbounded V-potential (abstract and §3) must be presented in full; the transition probabilities induced by the discretized linear potential should be written explicitly before the generating-function equation is solved, so that the absence of lattice artifacts can be verified directly.
  2. [§4] The asymptotic extraction of the logarithmic growth of the mean number of distinct sites visited (abstract) relies on the extreme-value cutoff k ~ log(t) from the geometric stationary measure; the manuscript should supply the explicit large-t expansion or Tauberian argument used to obtain this scaling.
minor comments (3)
  1. [§5] Notation for the bias strength and resetting probability should be unified across the V- and U-potential sections to avoid confusion when comparing the two cases.
  2. [Fig. 4] Figure captions for the mean-first-passage-time plots should state the precise initial and target site indices used, as the location dependence is emphasized in the text.
  3. [Introduction] A brief remark on the continuum limit of the lattice results would help readers connect the findings to the well-studied Brownian-motion cases mentioned in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We are pleased that the referee finds the results significant and agrees with the recommendation for minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] The central derivation of the generating function for the unbounded V-potential (abstract and §3) must be presented in full; the transition probabilities induced by the discretized linear potential should be written explicitly before the generating-function equation is solved, so that the absence of lattice artifacts can be verified directly.

    Authors: We agree with this suggestion. In the revised version, we will explicitly write the transition probabilities for the discretized V-potential at the start of §3, before deriving the generating function. This will make the discretization transparent and allow direct verification that no unintended lattice artifacts are introduced. revision: yes

  2. Referee: [§4] The asymptotic extraction of the logarithmic growth of the mean number of distinct sites visited (abstract) relies on the extreme-value cutoff k ~ log(t) from the geometric stationary measure; the manuscript should supply the explicit large-t expansion or Tauberian argument used to obtain this scaling.

    Authors: We appreciate this comment. The logarithmic scaling arises from the geometric decay of the stationary probability. We will add in the revised manuscript the explicit large-t asymptotic analysis, including the Tauberian theorem application or the direct expansion showing that the mean number of distinct sites S(t) ∼ (log t) / |log ρ| where ρ is the ratio of the geometric distribution, plus lower-order terms. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central derivations begin from the standard master equation for lattice random walks with position-dependent transition rates defined directly from the discretized potential differences. The generating function for occupation probabilities in the unbounded V-potential is obtained via standard techniques for birth-death chains, and the logarithmic growth of the mean number of distinct sites visited follows from the geometric decay of the stationary measure without any fitted parameters or self-referential definitions. First-passage quantities are extracted from the same generating functions or renewal equations, with no reduction to prior self-citations or ansatzes. The analysis of resetting and bounded cases similarly proceeds from explicit transition rules and does not invoke uniqueness theorems or imported results from the authors' prior work as load-bearing premises. All steps remain internally consistent and externally verifiable against the lattice master equation.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Markovian nearest-neighbor random-walk master equation modified by a position-dependent bias derived from the potential; no new entities are postulated.

free parameters (2)
  • bias strength
    The strength of the linear or quadratic bias is a tunable parameter whose value is not fixed by the derivation.
  • resetting probability
    The probability of resetting to a fixed site is introduced as a free control parameter.
axioms (2)
  • domain assumption The walker performs unbiased nearest-neighbor hops that are then reweighted by the Boltzmann factor of the potential at each step.
    Standard construction for lattice random walks in an external potential; invoked implicitly when defining the transition probabilities.
  • domain assumption The lattice is one-dimensional and translationally invariant except for the focal-point bias.
    Required for the generating-function approach described in the abstract.

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Reference graph

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