arxiv: 2605.08414 · v1 · submitted 2026-05-08 · ❄️ cond-mat.stat-mech

Recognition: 2 theorem links

· Lean Theorem

Mirror transitions in diffusion with stochastic resetting confined on a ring

Denis Boyer, Leonardo Dagdug, Pavel Castro-Villarreal, Pedro Juli\'an-Salgado

Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:04 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords diffusionstochastic resettingring geometrymean first-passage timeoptimal resetting ratephase transitionsmirror symmetrysurvival probability

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The pith

Optimal resetting rate for diffusion on a ring with two sites shows first-order jumps and second-order continuous changes as arc length to target and site weight vary.

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

The paper shows that a diffusing particle on a circular ring, reset stochastically to two sites and absorbed at one fixed target, has a mean first-passage time whose minimum with respect to the resetting rate can change character abruptly or smoothly. This occurs when the arc distance from one resetting site to the target or the probability weight assigned to the other site is varied. The resulting phase diagram in parameter space contains critical and tri-critical points, and every transition respects mirror symmetry about the target and the point directly opposite it on the ring. A reader would care because the result identifies concrete conditions under which the best resetting strategy for fastest arrival switches from one regime to another in a confined geometry.

Core claim

For a particle diffusing on a ring under stochastic resetting to two sites drawn from an arbitrary probability density and absorbed at a fixed target, the Laplace transform of the survival probability is obtained from the free propagator. The mean first-passage time is then minimized with respect to the resetting rate. Depending on the arc length between one resetting site and the target together with the weight of the remaining site, this optimal rate undergoes first-order (discontinuous) or second-order (continuous) transitions; the mean first-passage time itself admits critical and tri-critical points, and all such transitions are symmetric under reflection through the target and its diam

What carries the argument

Laplace transform of the survival probability computed from the free propagator of diffusion on the ring with resetting to an arbitrary density, used to obtain and optimize the mean first-passage time.

If this is right

The mean first-passage time surface in the two-site parameter space contains both critical and tri-critical points.
All transitions in the optimal resetting rate are mirror-symmetric about the target site and the diametrically opposite point.
For certain ranges of arc length and site weight the optimal rate changes continuously; for others it jumps abruptly.
The same symmetry and transition structure appears for any choice of target location on the ring.

Where Pith is reading between the lines

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

The same symmetry argument could be used to reduce the computational cost of optimizing resetting in multi-site or continuous-density cases on the ring.
Analogous first- and second-order transitions may appear when the ring is replaced by a finite interval with reflecting boundaries.
The existence of tri-critical points suggests that small changes in resetting protocol could switch the system between three qualitatively different optimization regimes.

Load-bearing premise

The calculations assume that a suitable free propagator exists for diffusion on the ring with resetting to sites drawn from any probability density and that the Laplace transform of the survival probability directly yields the mean first-passage time without further boundary or non-Markovian corrections.

What would settle it

A direct Monte Carlo simulation of Brownian motion on the ring with absorbing target and Poissonian resetting to two sites, for arc lengths near the predicted critical value, that checks whether the optimal resetting rate indeed jumps discontinuously or varies continuously.

Figures

Figures reproduced from arXiv: 2605.08414 by Denis Boyer, Leonardo Dagdug, Pavel Castro-Villarreal, Pedro Juli\'an-Salgado.

**Figure 2.** Figure 2: In white, we represent the pair values ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: System configuration for a Brownian particle hav [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: AMFPT in Eq. (33). In this no-resetting limit φ1 and φ2 are the possible initial sites of the free Brownian particle. The target and the first resetting site are fixed at φ∗ = π/2 and φ1 = 3π/2, respectively. φ1 = 3π 2 , allowing φ2 to vary over the entire interval [0, 2π]. Implementing these values into Eq. (30) we have ⟨Tr⟩ = 1 r    (m + 1) cosh ξrπ 1 + m cosh ξr π − [PITH_FULL_IMAGE:figures/full_f… view at source ↗

**Figure 5.** Figure 5: AMFPT as a function of r. Each panel represents a fixed value of m and each curve within a panel, a fixed value of φ2 (same color code). All φ2 values belong to the range φ2 ∈ [0, π/2]. For clarity, the inset of panel c) recalls the system under consideration, also shown in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Optimal parameter r ∗ vs φ2, defined by the expression in (40) for the AMFPT given by Eq. (31), for various values of m. Panel b) is a close-up view of panel a) near the tri-critical points (circular black symbols). The dashed vertical lines in panel b) show the discontinuous transitions. The jump magnitude of r ∗ becomes progressively larger as φ2t approaches π/2 (m decreases). See text for other details.… view at source ↗

**Figure 7.** Figure 7: Phase diagram of the case φ1 = 3π/2 in the (m, φ2)- plane: in the shaded area r ∗ ̸= 0, whereas r ∗ = 0 outside. On the solid lines, r ∗ reaches 0 continuously, while each point on the dashed lines represents a discontinuous transition where r ∗ abruptly drops to zero. The symmetry of the AMFPT described in Eq. (35) can now be directly appreciated in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 9.** Figure 9: “Order parameter” diagram r ∗ as a function of φ2, for a fixed φ1 = 5π/6 and several values of m. The inset provides a close-up of the transition region to the right of π/2 indicated by the black arrow, revealing the discontinuous behavior of r ∗ near the critical line (in red). The critical point is shown with the black dot. A pair of tri-critical points is marked by blue dots, located symmetrically arou… view at source ↗

**Figure 10.** Figure 10: Optimal resetting rate as a function of φ2 for the case φ1 = 2π/3 and several values of m. As in the previous case φ1 = 5π/6, this configuration exhibits a rich phenomenology. The inset provides a magnified view of the region indicated by the black arrow in the main panel, with the critical point highlighted in black. A second pair of critical points, located symmetrically around 3π/2, is indicated by … view at source ↗

read the original abstract

Diffusion with an incorporated resetting mechanism provides a reference framework for modeling a wide range of natural phenomena. Within this framework, the optimal resetting rate is a key quantity that arises from the optimization of the mean first-passage time. While substantial work has focused on the study of the optimal resetting rate in unbounded one dimensional domains, little is still known about the optimization of the mean first-passage time in bounded systems, in particular when multiple resetting sites are available. In this work, we consider a particle diffusing along a circular circumference and under resetting, with an absorbing target site at a fixed location. Using the appropriate free propagator for this system, we compute the Laplace transform of the survival probability when resetting occurs to multiple sites drawn from an arbitrary probability density function. We also calculate the mean first-passage time at the target site, and study the dependence of the optimal resetting rate in terms of the relevant parameters of the system in a two-resetting site configuration. Depending on the arc length between one of the resetting sites and the absorbing target site, and the weight of the remaining resetting site, the optimal resetting rate can exhibit abrupt ("first order'') and continuous ("second order'') transitions. Moreover, the behavior of the mean first-passage time is rich enough to allow both critical and tri-critical points to exist in the parameter space. All the transitions have "mirror symmetry'' around the selected target site and its corresponding diametrically opposite site.

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

This paper shows first- and second-order transitions in the optimal resetting rate for diffusion on a ring with two sites, plus critical points and mirror symmetry around the target.

read the letter

The core result is that when a diffusing particle on a ring resets to two fixed sites with adjustable weights, the resetting rate that minimizes the mean first-passage time to an absorbing target can switch abruptly or vary smoothly as the arc length or site weights change. These transitions come with mirror symmetry around the target and its diametric opposite, and the MFPT surface supports both critical and tri-critical points in parameter space. That behavior is new compared with the usual unbounded-line resetting studies. The authors derive it from the free propagator that includes diffusion plus Poissonian resetting to multiple sites, take the Laplace transform of the survival probability, extract the MFPT in the s to 0 limit, and then minimize with respect to the rate r. The construction stays Markovian and respects the ring periodicity, so the optimization landscape is well-defined. The first-order jumps occur when the global minimum switches between two local minima; the second-order cases appear when a minimum moves through r=0 or merges with another extremum. This matches the stress-test description and looks internally consistent. The two-site restriction is a clean starting point that already produces rich structure. Soft spots are limited. The abstract and outline give the steps but no explicit derivations or error bars appear in the provided material, so a referee would want to see the full algebra and at least some numerical checks that the analytic minima match simulated first-passage times. Generalization beyond two sites is mentioned but not explored in depth. The work is aimed at people who already follow resetting diffusion or search processes in bounded domains. It is a straightforward extension that adds concrete new phenomena without forcing extra assumptions. I would send it to peer review. The central construction holds up and the reported transitions are worth checking in detail.

Referee Report

2 major / 3 minor

Summary. The manuscript studies diffusion on a ring with stochastic resetting to multiple sites (focusing on two sites) and an absorbing target. It derives the Laplace transform of the survival probability from the free propagator incorporating diffusion and Poissonian resetting, computes the mean first-passage time (MFPT) via the s→0 limit, and minimizes the MFPT with respect to the resetting rate r. The central results are that the optimal r exhibits first-order (abrupt) and second-order (continuous) transitions depending on arc length between a resetting site and target and the weight of the second site; the MFPT landscape permits critical and tri-critical points; and all transitions display mirror symmetry around the target and its diametrically opposite point.

Significance. If the derivations are correct, the work meaningfully extends resetting optimization studies from unbounded domains to bounded circular geometries with multiple sites. The analytical identification of mirror-symmetric first- and second-order transitions in the optimal resetting rate, together with the existence of critical and tri-critical points, supplies a rich, falsifiable phase diagram that can benchmark simulations and inform models in statistical mechanics and stochastic processes. The use of free propagators and Laplace transforms for exact MFPT expressions is a methodological strength.

major comments (2)

[two-resetting-site configuration] The explicit minimization of MFPT(r) and the condition for the global minimum switching between distinct local minima (producing first-order transitions) must be shown in the two-site analysis section; without this, it is unclear whether the reported transitions follow directly from the Laplace-transformed survival probability or require additional assumptions.
[derivation of Laplace transform] The free propagator for diffusion plus resetting to an arbitrary density on the ring (used to obtain the Laplace transform of the survival probability) should be stated explicitly, including how ring periodicity and the absorbing target are incorporated via the renewal relation; this is load-bearing for all subsequent claims about transitions.

minor comments (3)

[figures] Figure captions for plots of MFPT versus r should explicitly list the arc-length and weight values used to demonstrate the first- and second-order transitions.
[introduction] The introduction would benefit from additional citations to prior work on resetting in bounded domains to sharpen the novelty claim.
[methods] Notation for the resetting-site probability density function should be unified between the general multi-site case and the two-site specialization to avoid ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment of its significance. We address each major comment below and will make the requested clarifications in the revised version.

read point-by-point responses

Referee: [two-resetting-site configuration] The explicit minimization of MFPT(r) and the condition for the global minimum switching between distinct local minima (producing first-order transitions) must be shown in the two-site analysis section; without this, it is unclear whether the reported transitions follow directly from the Laplace-transformed survival probability or require additional assumptions.

Authors: We agree that an explicit demonstration of the minimization would clarify the origin of the transitions. The MFPT is obtained directly as the s→0 limit of the Laplace transform of the survival probability derived from the free propagator. In the revised manuscript we will add, in the two-site analysis section, the explicit steps: computation of d(MFPT)/dr = 0 to locate local minima, followed by direct comparison of MFPT values at those competing minima to determine the parameter values at which the global minimum switches discontinuously. This will show that the first-order transitions arise strictly from the MFPT expression without additional assumptions. revision: yes
Referee: [derivation of Laplace transform] The free propagator for diffusion plus resetting to an arbitrary density on the ring (used to obtain the Laplace transform of the survival probability) should be stated explicitly, including how ring periodicity and the absorbing target are incorporated via the renewal relation; this is load-bearing for all subsequent claims about transitions.

Authors: We thank the referee for highlighting this point. Although the manuscript invokes the appropriate free propagator and applies the renewal relation, we will state the propagator explicitly in the revised version. The expression will be written for diffusion plus Poissonian resetting to an arbitrary density on the ring, with periodicity enforced via summation over images (or equivalent Fourier representation on the circle). The renewal relation will be displayed in full, showing how the Laplace transform of the survival probability is constructed from the free propagator, with the absorbing target incorporated through the first-passage structure of the integral equation. This will make the load-bearing derivation transparent and support all subsequent results on the transitions. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained; no circular reductions identified

full rationale

The central derivation begins with the free propagator for diffusion on a ring with Poissonian resetting to arbitrary sites, computes its Laplace transform to obtain the survival probability, extracts the MFPT via the s→0 limit, and minimizes the resulting expression with respect to the resetting rate r. The first- and second-order transitions, critical points, and mirror symmetry emerge directly from the analytic form of this MFPT(r) landscape as the arc length and site weights are varied; no parameter is fitted to data and then relabeled as a prediction, no load-bearing premise rests on a self-citation, and no ansatz or uniqueness claim is smuggled in. The construction follows standard renewal relations for Markovian first-passage problems and remains internally consistent without reducing to its own inputs by definition.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The model rests on standard assumptions of Brownian motion with instantaneous resetting and absorbing boundaries on a ring; no new entities are postulated and the only free parameters are the physical variables (reset rate, site positions, weights) that are varied to locate optima.

free parameters (2)

resetting rate
Varied continuously to locate the minimum of the mean first-passage time.
reset site weights
Probability weights for the two sites treated as tunable parameters in the two-site case.

axioms (1)

domain assumption Standard Markovian diffusion on a ring with absorbing target and Poissonian resetting to sites drawn from a probability density.
Invoked when stating the use of the free propagator and Laplace transform of the survival probability.

pith-pipeline@v0.9.0 · 5568 in / 1433 out tokens · 66934 ms · 2026-05-12T02:04:03.725552+00:00 · methodology

discussion (0)

Reference graph

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