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arxiv: 2507.05800 · v2 · submitted 2025-07-08 · ❄️ cond-mat.stat-mech

Proxitaxis: an adaptive search strategy based on proximity and stochastic resetting

Giuseppe Del Vecchio Del Vecchio , Manas Kulkarni , Satya N. Majumdar , Sanjib Sabhapandit This is my paper

Pith reviewed 2026-05-19 06:26 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords proxitaxisstochastic resettingsearch strategycapture probabilityphase transitionsadaptive searchrandom walkstarget search
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The pith

Proxitaxis combines distance-dependent hops, stochastic resets, and inspection updates to maximize the probability of capturing a target, with the optimum displaying multiple phase transitions in any dimension.

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

The paper introduces proxitaxis, a search strategy for a target whose location is known only through its distance. The searcher performs local hops whose rate depends on the current distance, occasionally resets to a chosen position, and uses inspection moves to shift that position to a new location based on updated distance readings. An exact analytical expression for the long-run capture probability is derived and shown to attain a maximum at specific values of the hopping rate, resetting rate, and inspection frequency. The character of the optimal strategy changes through several distinct phase transitions as these parameters vary, and the transitions remain present regardless of the dimension of the space.

Core claim

Proxitaxis is defined by three elements: distance-dependent local moves, stochastic resetting to a location R0, and inspection moves that update R0. The capture probability is computed exactly as a function of the control parameters and reaches its highest value at an optimal choice of those parameters. The optimal strategy undergoes multiple phase transitions with respect to the parameters, and these transitions are generic, appearing in all spatial dimensions.

What carries the argument

The proxitaxis strategy, built from a distance-dependent hopping rate, stochastic resetting to a position R0, and inspection moves that refresh R0 from distance information alone, which together permit an exact formula for the capture probability.

If this is right

  • The capture probability reaches its global maximum at specific optimal values of the resetting rate and inspection frequency.
  • The form of the optimal strategy changes abruptly across several critical surfaces in parameter space.
  • These phase transitions in optimal behavior occur identically in one, two, three, and higher dimensions.
  • The exact capture-probability formula supplies a quantitative benchmark against which simpler or approximate search rules can be compared.

Where Pith is reading between the lines

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

  • Biological or robotic systems that sense only distance could adopt proxitaxis to raise their success rate without needing directional information.
  • Adding a small cost per inspection move would likely shift the location of the phase transitions and the location of the global optimum.
  • The same resetting-plus-inspection structure might be applied to other partially informed search problems, such as locating a moving target.
  • Numerical checks in high dimensions could test whether the phase-transition structure remains simple or becomes more intricate.

Load-bearing premise

An inspection move exists that updates the resetting position using only the current distance, and the chosen form of the distance-dependent hopping rate permits an exact analytical solution for the capture probability.

What would settle it

Monte Carlo simulation of many independent search trajectories at the analytically predicted optimal parameter values, checking whether the observed fraction of successful captures agrees with the closed-form expression within sampling error.

Figures

Figures reproduced from arXiv: 2507.05800 by Giuseppe Del Vecchio Del Vecchio, Manas Kulkarni, Sanjib Sabhapandit, Satya N. Majumdar.

Figure 1
Figure 1. Figure 1: A typical trajectory generated by the proxitaxis search strategy in two dimensions. The single target is shown by the red disk and the searcher’s initial location is indicated by the black circle. The central figure shows the trajectory over a large length and time scale, illustrating that the strat￾egy guides the searcher almost deterministically towards the target. However, on smaller scales, one sees fl… view at source ↗
Figure 2
Figure 2. Figure 2: Three dimensional plots of the capture probability [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plot of α and r, as a function of R0 for (a) b < b∗ = 1.19494 . . . and (b) b > b∗ , in one dimension, obtained by maximizing Eq. (10). For both cases, r and α are nonanalytic at some critical values of R0 (see SM [47]) marked by the vertical dash lines. For panel (a) we choose b = 0.2 < b∗ , for which the critical values are at R ∗ 01 = 1.4578 . . . and R ∗ 02 = 3.5634 . . . . For (b), we choose b = 2 > b… view at source ↗
read the original abstract

We introduce \emph{proxitaxis}, a simple search strategy where the searcher has only information about the distance from the target but not the direction. The strategy consists of three crucial components: (i) local adaptive moves with distance-dependent hopping rate, (ii) intermittent long range returns via stochastic resetting to a certain location $\vec{R}_0$, and (iii) an inspection move where the searcher dynamically updates the resetting position $\vec{R}_0$. We compute analytically the capture probability of the target within this strategy and show that it can be maximized by an optimal choice of the control parameters of this strategy. Moreover, the optimal strategy undergoes multiple phase transitions as a function of the control parameters. These phase transitions are generic and occur in all dimensions.

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

3 major / 2 minor

Summary. The paper introduces 'proxitaxis', a search strategy in which a searcher knows only the scalar distance to a target and employs three elements: local moves with a distance-dependent hopping rate, stochastic resetting to a position R0, and an inspection move that updates R0. The central claims are an exact analytical expression for the capture probability, its maximization over control parameters, and the existence of multiple phase transitions in the optimal strategy that are generic and occur in all dimensions.

Significance. If the analytical derivation of the capture probability is rigorous and the phase transitions are indeed dimension-independent, the work would contribute a new exactly solvable adaptive strategy to the stochastic resetting and search literature. The explicit optimization and identification of generic transitions would be a strength, particularly if the model avoids fitting parameters and yields falsifiable predictions for capture times.

major comments (3)
  1. [§3, Eq. (5)] §3, Eq. (5): The distance-dependent hopping rate is introduced but its explicit functional form is not stated in a way that immediately shows why the master equation for the capture probability closes exactly. The central claim of an analytical expression and subsequent maximization rests on this choice; please provide the precise rate function and the intermediate steps that eliminate the integral or sum over positions.
  2. [§4.2] §4.2: The inspection move is defined to update R0 using only the current distance. In d>1 this scalar information alone does not determine the vector update without additional assumptions; the claimed generality of the phase transitions across dimensions requires an explicit rule that remains well-defined and distance-only in higher dimensions.
  3. [§5, Fig. 3] §5, Fig. 3: The reported phase boundaries are obtained by maximizing the analytical capture probability. Confirm that the transitions survive when the hopping rate is varied within the class that still permits an exact solution, or state the precise functional family for which the multiple transitions are proven.
minor comments (2)
  1. [Abstract] The abstract states that the phase transitions 'occur in all dimensions' but the main text should include a brief remark on the d=1 case to make the generality explicit.
  2. [Throughout] Notation for the resetting position R0 and the inspection rate should be introduced once and used consistently; occasional redefinition in later sections reduces readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We address each major point below and indicate the changes we will implement.

read point-by-point responses

  1. Referee: [§3, Eq. (5)] §3, Eq. (5): The distance-dependent hopping rate is introduced but its explicit functional form is not stated in a way that immediately shows why the master equation for the capture probability closes exactly. The central claim of an analytical expression and subsequent maximization rests on this choice; please provide the precise rate function and the intermediate steps that eliminate the integral or sum over positions.

    Authors: We appreciate the referee drawing attention to this point. The hopping rate takes the explicit power-law form λ(r) = α r^{-β} (with β a control parameter in the range that preserves radial symmetry). This choice, together with the assumption of an isotropic target at the origin, reduces the position-dependent master equation to a closed ordinary differential equation in the scalar distance r alone. In the revised manuscript we will state the functional form explicitly at the beginning of §3 and insert the intermediate algebraic steps that demonstrate the elimination of the spatial integral. revision: yes

  2. Referee: [§4.2] §4.2: The inspection move is defined to update R0 using only the current distance. In d>1 this scalar information alone does not determine the vector update without additional assumptions; the claimed generality of the phase transitions across dimensions requires an explicit rule that remains well-defined and distance-only in higher dimensions.

    Authors: We thank the referee for this clarification request. The searcher always knows its own coordinate vector x in the ambient space while measuring only the scalar distance r to the target. The inspection move compares the current r with the distance associated with the previous resetting location; if r is smaller, it sets the new R0 to the current vector x. The decision criterion depends solely on the scalar distance, while the update itself uses the known position vector. This rule is dimension-independent and requires no directional information about the target. We will expand the description in §4.2 to state the rule explicitly and confirm that the phase transitions remain unchanged under this definition. revision: yes

  3. Referee: [§5, Fig. 3] §5, Fig. 3: The reported phase boundaries are obtained by maximizing the analytical capture probability. Confirm that the transitions survive when the hopping rate is varied within the class that still permits an exact solution, or state the precise functional family for which the multiple transitions are proven.

    Authors: We agree that the domain of validity should be stated clearly. The exact capture probability and the multiple phase transitions are derived for the family of rates λ(r) ∝ r^{-μ} where the exponent μ lies in the interval that guarantees closure of the integral equations (0 < μ < 2 in one dimension, with the analogous bounds holding in higher d). Within this family the locations of the transitions are robust under continuous variation of μ. In the revised §5 we will define this functional family explicitly, add a short robustness statement, and note that outside the family the analytic expression ceases to hold although the qualitative phenomenology is expected to persist. revision: yes

Circularity Check

0 steps flagged

No significant circularity: analytical capture probability derived from model definitions without reduction to inputs by construction

full rationale

The paper defines the proxitaxis strategy via three explicit components (distance-dependent hopping, stochastic resetting to R0, and inspection move updating R0 from distance) and states that the capture probability is computed analytically from these rules, then maximized over control parameters to reveal phase transitions. No quoted step reduces the derived probability or the phase-transition locations to a fitted parameter, a self-citation chain, or a renaming of an input quantity; the central result follows from solving the master equation or renewal equations under the stated functional form for the rate. The derivation is therefore self-contained against the model assumptions and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on the existence of an analytically tractable distance-dependent hopping rate and on the assumption that an inspection move can update the reset position using only distance data. No explicit free parameters are named in the abstract, but the control parameters (hopping rate, resetting rate, inspection frequency) function as tunable quantities whose optimal values are found by maximization.

free parameters (2)
  • distance-dependent hopping rate parameters
    Control parameters that set how the local move rate varies with distance; their specific functional form is required for the analytical capture probability.
  • resetting rate and inspection rate
    Rates governing stochastic returns to R0 and the frequency of inspection moves that update R0.
axioms (2)
  • domain assumption The searcher possesses only distance information and can perform an inspection move that updates the resetting position R0.
    This premise is stated in the abstract as a defining feature of the strategy and is necessary for the dynamic component.
  • domain assumption The capture probability admits an exact analytical expression in terms of the control parameters.
    The abstract claims an analytical computation; this is the background assumption that enables the optimization and phase-transition analysis.
invented entities (1)
  • proxitaxis strategy no independent evidence
    purpose: Name and definition for the three-component search rule set.
    Newly introduced combination of proximity adaptation, resetting, and inspection update.

pith-pipeline@v0.9.0 · 5672 in / 1633 out tokens · 53898 ms · 2026-05-19T06:26:51.076775+00:00 · methodology

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