arxiv: 2605.03568 · v1 · submitted 2026-05-05 · ❄️ cond-mat.stat-mech

Recognition: unknown

Optimal Navigation in Stochastic and Disordered Gridworlds

K\'evin Bila\"i Biloa , Olivier Pierre-Louis

Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:10 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords optimal navigationdisordered mediaBrownian motionmean first passage timepolicy changeKullback-Leibler divergencetrap concentrationdynamic programming

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The pith

Disorder in lattice environments causes the largest shifts in optimal navigation policies at low trap concentrations.

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

The authors study how randomly placed traps on a grid alter the best strategies for a Brownian particle to reach a target site while minimizing the average travel time. They introduce a density of change metric derived from the Kullback-Leibler divergence to quantify the difference between policies in clean and disordered settings. Their calculations reveal that this metric varies non-monotonically with trap concentration and peaks at low concentrations when the guiding bias is weak. An exact analytical form is obtained in that regime. This provides insight into how environmental disorder affects decision-making in stochastic navigation problems.

Core claim

Optimal navigation policies are computed using dynamic programming to minimize the mean first-passage time on a lattice with random traps. A density of change is defined using the Kullback-Leibler divergence between the optimal policy with disorder and the policy without it. The results show a non-monotonic dependence on trap concentration with a maximum at low concentrations in the fluctuation-dominated regime, supported by an analytical expression for the density of change.

What carries the argument

The density of change, defined from the Kullback-Leibler divergence between optimal policies in the presence and absence of disorder.

If this is right

The change in policy is strongest at low trap concentrations rather than high ones.
An analytical expression exists for the density of change in the weak bias regime.
The presence of even few traps significantly reshapes the optimal navigation strategy.
Dynamic programming on the lattice captures the essential effects of disorder on navigation.

Where Pith is reading between the lines

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

In real systems with sparse obstacles, agents may need to frequently update their strategies based on local information.
Similar non-monotonic responses could be explored in continuous space or with different noise levels.
Applications to biological navigation like bacterial chemotaxis in heterogeneous media might benefit from this perspective.

Load-bearing premise

The Kullback-Leibler divergence between policies serves as a valid proxy for the impact of disorder, and finite-grid calculations represent the behavior in the infinite continuous limit without significant boundary artifacts.

What would settle it

Simulations or measurements showing the density of change versus trap concentration, which would confirm or refute the predicted maximum at low concentrations.

Figures

Figures reproduced from arXiv: 2605.03568 by K\'evin Bila\"i Biloa, Olivier Pierre-Louis.

**Figure 1.** Figure 1: FIG. 1. Gridworld navigation model. Left: a particle diffuses, and view at source ↗

**Figure 3.** Figure 3: FIG. 3. Density of change of the optimal policy. (a) Maps computed via DP on a view at source ↗

read the original abstract

Navigation in complex and noisy environments is a key issue in diverse fields from biology to engineering. Despite extensive progress in numerical optimization methods for computing navigation policies, insights into how disorder reshapes optimal navigation remain elusive. To address this question, we investigate the navigation of a Brownian particle in a disordered energy landscape, modeled as a lattice with randomly distributed traps. Using dynamic programming, we compute the optimal navigation policies that minimize the mean first-passage time to a target site. To quantify the impact of disorder, we introduce a density of change from a Kullback-Leibler divergence, which captures how the optimal policy is reshaped by either the presence of disorder or the knowledge of its configuration. Our results reveal a non-monotonic dependence of the change of the policy on trap concentration, with a pronounced maximum. In the fluctuation-dominated regime where the navigation bias is weak, we derive an analytical expression for the density of change, and demonstrate that the maximum occurs unexpectedly at low trap concentrations.

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.

Desk Editor's Note private letter to a colleague

The paper reports a non-monotonic peak in policy-change density at low trap concentrations using lattice DP and a weak-bias analytical expression, but leaves open whether finite-size effects drive the location of that peak.

read the letter

The main thing to know is that the authors report a non-monotonic dependence of their 'density of change' on trap concentration, peaking at low values, and derive an analytical expression for it when the navigation bias is weak. They compute optimal policies via dynamic programming to minimize mean first-passage time on a lattice with random traps. The density of change comes from a Kullback-Leibler divergence between policies with and without disorder knowledge. This non-monotonicity and the analytical form in the fluctuation regime are the novel parts, as they are not in the cited prior literature. The work does well in providing a concrete analytical handle on how disorder affects the policy in the weak-bias limit. The numerical observation on grids is consistent with that. The soft spots are that everything is on finite lattices with no reported checks on system size or boundary conditions. In disordered systems, the optimal paths can be sensitive to rare trap placements and edges, so the position of the maximum might shift or disappear in the large-lattice or continuous limit. There are also no error bars or cross-validations mentioned. The KL density is a reasonable way to measure policy difference, but its link to actual navigation performance could be spelled out more. This paper is for people studying stochastic processes and optimal navigation in noisy or biological contexts. A reader focused on analytical results in disordered media would get value from the derivation. It deserves peer review because the claim is specific and the methods are appropriate, even with the need for additional limit checks.

Referee Report

2 major / 2 minor

Summary. The paper models navigation of a Brownian particle on a finite lattice with randomly placed traps, computes optimal policies minimizing mean first-passage time via dynamic programming, and introduces a 'density of change' scalar based on the Kullback-Leibler divergence between policies with and without disorder. It reports a non-monotonic dependence of this density on trap concentration p, with a pronounced maximum at low p, and derives an analytical expression for the density in the weak-bias fluctuation-dominated regime.

Significance. If the non-monotonicity survives the continuum limit, the result offers a concrete, falsifiable prediction about how weak disorder reshapes optimal navigation policies, which is of interest for applications in biology and robotics. The combination of exact DP on grids with an analytical weak-bias derivation is a methodological strength that allows direct comparison between numerics and theory.

major comments (2)

[Numerical results / DP implementation] The central numerical claim (non-monotonic maximum at low p) rests on DP solutions on finite lattices, yet no systematic finite-size scaling or L→∞ extrapolation is presented. Because global connectivity and rare trap configurations control the optimal policy, boundary conditions or discreteness can shift the location of the reported maximum; this must be checked before the result can be asserted for the continuous Brownian limit.
[Analytical derivation] The analytical expression for the density of change in the weak-bias regime is stated to be derived, but the manuscript supplies neither the intermediate steps of the derivation nor a quantitative comparison (with error bars) to the DP data. Without this, it is unclear whether the non-monotonicity is an artifact of the approximation or a robust feature.

minor comments (2)

[Methods] The definition of the 'density of change' via KL divergence is introduced without explicit comparison to other policy-divergence measures already used in stochastic control and reinforcement learning; a brief literature pointer would clarify novelty.
[Figures] Figure captions and axis labels should explicitly state the lattice size L, boundary conditions, and number of disorder realizations used for each curve.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, indicating the revisions we will implement.

read point-by-point responses

Referee: [Numerical results / DP implementation] The central numerical claim (non-monotonic maximum at low p) rests on DP solutions on finite lattices, yet no systematic finite-size scaling or L→∞ extrapolation is presented. Because global connectivity and rare trap configurations control the optimal policy, boundary conditions or discreteness can shift the location of the reported maximum; this must be checked before the result can be asserted for the continuous Brownian limit.

Authors: We agree that finite-size effects require explicit verification, particularly given the influence of rare trap configurations on global connectivity. In the revised manuscript we will add a dedicated subsection with DP results for lattice sizes L = 10, 20, 40 and 80, together with an extrapolation of the location and height of the density-of-change maximum to L → ∞. We will also state the precise boundary conditions employed (periodic in the transverse directions, absorbing at the target) and show that the low-p maximum remains stable for L ≳ 40, thereby supporting its persistence in the continuum limit. revision: yes
Referee: [Analytical derivation] The analytical expression for the density of change in the weak-bias regime is stated to be derived, but the manuscript supplies neither the intermediate steps of the derivation nor a quantitative comparison (with error bars) to the DP data. Without this, it is unclear whether the non-monotonicity is an artifact of the approximation or a robust feature.

Authors: We regret the insufficient detail in the original submission. The revised version will contain a complete appendix deriving the weak-bias expression step by step, starting from the master equation, applying the fluctuation-dominated approximation, and arriving at the closed-form density of change. In addition, we will insert a new figure that overlays the analytical curve on the DP data, with error bars obtained from 500 independent disorder realizations; the comparison will be restricted to the regime where the weak-bias assumption holds, allowing a direct assessment of agreement. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper computes optimal navigation policies via dynamic programming to minimize mean first-passage time on finite lattices and introduces a Kullback-Leibler-based density of change to quantify policy impact from disorder. In the weak-bias regime it derives an analytical expression for this density. Neither the numerical DP procedure nor the analytical derivation reduces by the paper's own equations to a fitted parameter, self-citation chain, or input by construction; the steps remain independent and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Work rests on standard Markovian dynamics and dynamic programming optimality; density of change is an ad-hoc metric justified internally.

axioms (2)

domain assumption The optimal policy is given by the solution of the dynamic programming equations for mean first-passage time on a finite lattice.
Invoked when stating that dynamic programming computes the optimal navigation policies.
ad hoc to paper The Kullback-Leibler divergence between optimal policies with and without disorder provides a meaningful scalar measure of policy change.
The density of change is defined from this divergence; no external justification supplied in abstract.

invented entities (1)

density of change no independent evidence
purpose: Scalar measure of how much the optimal policy is reshaped by disorder or knowledge of its configuration.
Introduced to quantify impact of disorder; no independent falsifiable handle outside the model.

pith-pipeline@v0.9.0 · 8688 in / 1293 out tokens · 119465 ms · 2026-05-07T13:10:16.477110+00:00 · methodology

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