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Exact collective first-passage statistics of N trail-interacting walkers

This paper derives exact expressions for collective first-passage statistics of $N$ trail-interacting walkers, revealing that splitting probabilities remain invariant whether walkers move simultaneously or sequentially, provided their response to the shared environmental memory is saturating.

Original authors: Paul Pineau, Julien Brémont, Olivier Bénichou, Raphaël Voituriez

Published 2026-07-16

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CC BY 4.0

Original authors: Paul Pineau, Julien Brémont, Olivier Bénichou, Raphaël Voituriez

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded dance floor where everyone is trying to find the exit, but the floor itself is changing under their feet. This is the world of "active matter," a branch of physics that studies how living things (like bacteria or ants) or synthetic robots move and interact. Usually, scientists think of these particles as independent dancers, each following their own rhythm. But in reality, many creatures leave behind a "trail"—a chemical scent, a physical mark, or a memory in the environment—that tells others where they've been. Think of it like a hiker leaving a line of pebbles; the next hiker sees the pebbles and knows which path was taken before. When many walkers share this same trail, they aren't just moving through space; they are moving through a shared history. The big question scientists have been asking is: how does this shared memory change the odds of who finds the exit first, or who gets stuck? It's a puzzle about how the past shapes the future for a whole group, not just a single traveler.

In this new study, a team of physicists decided to crack this puzzle by imagining a group of $N$ walkers on a one-dimensional line, like beads on a string, all leaving and following the same trail. They wanted to know two specific things: first, how long does it take for the $k$ -th walker to hit a target (like a wall at the end of the line)? And second, if there are two exits (a wall on the left and a wall on the right), what are the chances that exactly $k$ walkers will choose the left exit while the rest go right?

The researchers found something truly surprising, but only for a specific type of walker. They focused on "saturating" walkers—creatures that get tired of following the trail. Imagine a dog sniffing a scent; at first, the smell is strong and it follows it eagerly. But if the scent is everywhere, the dog eventually gets bored and stops paying attention. Mathematically, this means the trail's influence hits a "ceiling" and stops getting stronger no matter how many times it's crossed. For these specific walkers, the team discovered a magical rule: it doesn't matter if the walkers start all at once (a simultaneous start) or one by one (a sequential start). Whether they are a chaotic crowd rushing out together or a polite line of people waiting their turn, the odds of who ends up at which wall are exactly identical. The shared memory of the trail somehow "washes out" the difference between starting together or starting apart. They derived exact formulas for these probabilities and for how long it takes the walkers to get absorbed, showing that the group behaves with a mathematical elegance that defies the usual chaos of history-dependent systems.

However, this magic trick has a limit. The researchers also looked at "non-saturating" walkers—creatures that never get bored, where the trail gets infinitely stronger the more it is crossed (like a snowball rolling down a hill, getting bigger and bigger). For these walkers, the story changes completely. In this case, the starting method does matter. If they start one by one, the first walker leaves a massive, overwhelming trail that the second walker is forced to follow, changing the outcome compared to if they had all started together. The "invariance" (the rule that the outcome is the same regardless of the start time) breaks down.

To reach these conclusions, the team used a clever mathematical trick involving something called "squared Bessel processes," which are fancy ways of describing how random paths grow and interact. They proved that for the saturating walkers, the total "local time" (the amount of time spent on each spot) created by a group starting together is statistically the same as the sum of the local times created by walkers starting one after another. They backed up their math with massive computer simulations, running millions of virtual experiments to confirm that their formulas matched the digital reality. While they couldn't prove the exact same formulas for the non-saturating walkers (because the math gets too messy), their simulations clearly showed that the two starting protocols produce different results for those types of walkers.

So, the main takeaway is a beautiful distinction in the physics of memory: if the memory of the trail has a limit (it saturates), the group's fate is the same whether they move as a pack or a line. But if the memory grows without bound, the order in which they move changes everything. This work provides a precise toolkit for predicting how groups of active agents—whether they are ants, cells, or future swarms of robots—will explore their world when they are all leaving and reading the same footprints.

Technical Summary: Exact Collective First-Passage Statistics of N Trail-Interacting Walkers

Problem Statement
The paper addresses the collective first-passage statistics of $N$ one-dimensional random walkers interacting through a shared "trail" field. Unlike classical models where agents interact via instantaneous pairwise forces, these walkers modify their environment by depositing cues (trails) that persist over time. This creates a memory field that mediates interactions, leading to intrinsically non-Markovian and non-stationary dynamics. While single-particle self-interacting random walks (SIRWs) have been extensively studied, the quantitative characterization of collective phenomena—specifically first-passage times and splitting probabilities for $N$ interacting agents—remains largely unexplored. The authors aim to derive exact analytical expressions for these observables, focusing on how the history-dependent correlations generated by the common environment affect exploration efficiency and target competition.

Methodology
The authors analyze two broad classes of SIRWs defined by the asymptotic behavior of the weight function $w(n)$ , which determines the jump probability based on the local time (number of edge crossings) $L_t(x)$ :

Saturating SIRWs: The weight function saturates to a constant $l > 0$ as $n \to \infty$ . The primary model studied is the Once-Reinforced Random Walk (SATW $_\phi$ ), where $w(0) = 1/\phi$ and $w(n \ge 1) = 1$ .
Non-saturating SIRWs: The weight function decays to zero, either algebraically (PSRW) or exponentially (TSAW).

The core theoretical approach involves an exact representation of the collective trail field using Ray-Knight theorems generalized from single-particle theory (originally developed by Tóth). The authors map the local time process of the walkers to squared Bessel processes (BESQ).

For the saturating case (SATW $_\phi$ ), they establish that the mutual local-time profile after absorption admits an exact representation in terms of BESQ processes with specific dimensions dependent on $N$ and the parameter $\phi$ .
They utilize the additivity property of squared Bessel processes to demonstrate the equivalence between sequential (walkers launched one after another) and simultaneous (all walkers launched together) protocols for saturating interactions.
For non-saturating cases, where exact analytical constructions of the local time profile are unavailable, the authors rely on numerical simulations and scaling arguments.

Key Contributions and Results

Generalized Persistence Exponents:
The authors derive exact expressions for the persistence exponents $\theta_{k,N}$ , which characterize the long-time decay of the survival probability $S_{k,N}(t) \sim t^{-\theta_{k,N}}$ (the probability that at least $k$ out of $N$ walkers survive up to time $t$ ).
- For SATW $_\phi$ , the exact spectrum is given by:
  $\theta_{k,N}^{(\text{SATW}_\phi)} = \frac{k + \phi - 1}{2}$
- For generic SIRWs (including non-saturating classes), the exponent for the last surviving walker ( $k=1$ ) is found to be identical to the single-particle exponent:
  $\theta_{1,N}^{(\text{SIRW})} = \theta_{1,1}^{(\text{SIRW})}$
  This result implies that for the last survivor, the presence of other walkers does not alter the asymptotic decay rate, as the walker eventually explores uncorrelated territory.
Splitting Probabilities and Protocol Invariance:
The paper calculates the splitting probability $P(+k|x_0)$ , defined as the probability that exactly $k$ walkers are absorbed at one boundary (target 1) while $N-k$ are absorbed at the other (target 0), starting from $x_0 \in (0,1)$ .
- Exact Equivalence: A striking result for the saturating class (SATW $_\phi$ ) is that the splitting probabilities are exactly identical whether the walkers are launched simultaneously or sequentially. The authors prove this by showing that the cumulative local time profile is invariant under the launch protocol due to the additivity of the underlying BESQ processes.
- Analytical Formula: For SATW $_\phi$ , the splitting probability is given by:
  $P(+k|x_0) = \binom{N + 2(\phi - 1)}{k + \phi - 1} x_0^{k+\phi-1} (1-x_0)^{N-k+\phi-1}$
  (with appropriate generalization for $k=0$ and $k=N$ involving the incomplete Beta function).
- Breakdown for Non-Saturating Interactions: Numerical analysis demonstrates that this invariance breaks down for non-saturating interactions (e.g., TSAW). In these cases, the splitting probabilities depend on the launch protocol, with simultaneous launches effectively enhancing spatial spreading and repulsion compared to sequential launches.
Numerical Verification:
The theoretical predictions are validated through extensive numerical simulations on 1D lattices. The simulations confirm the persistence exponents for both saturating and non-saturating models and verify the protocol invariance for SATW $_\phi$ while highlighting the divergence for TSAW.

Significance and Claims
The authors claim to have established an exact analytical framework for collective first-passage observables in systems coupled through persistent environmental memory. The primary significance lies in:

Exact Solvability: Providing the first exact expressions for the full spectrum of persistence exponents and splitting probabilities for $N$ interacting walkers, a problem previously inaccessible to fluctuating hydrodynamics approaches.
Protocol Invariance: Revealing a deep invariance property where the statistics of competing targets remain unchanged between sequential and simultaneous exploration protocols, provided the trail interaction is saturating. This suggests that despite strong history-dependent correlations, the collective outcome is robust to the timing of agent introduction.
Universality: The results for SATW $_\phi$ are shown to extend to a broader class of saturating self-interacting walks via an effective parameter $\phi$ .
Biological and Synthetic Relevance: The framework offers quantitative predictions for first-passage phenomena in biological systems (e.g., cells modifying their environment, ant trail following) and synthetic active matter (e.g., active droplets depositing persistent trails), where memory-mediated interactions are known to enhance spatial spreading.

The paper concludes that while trail-mediated interactions generate strong non-Markovian dynamics, the collective first-passage statistics for saturating interactions can be mapped exactly to known stochastic processes (squared Bessel processes), allowing for precise predictions of exploration efficiency and target competition.

Technical Summary: Exact Collective First-Passage Statistics of N Trail-Interacting Walkers

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