Extreme First-Passage Time of Many Interacting Particles

arXiv:2607.22528 · cond-mat.stat-mech, cond-mat.mes-hall, cond-mat.soft, math-ph, math.MP, physics.bio-ph · Submitted 2026-07-24 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Hi, I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Extreme First-Passage Time of Many Interacting Particles".

Mira: Here's the gist of the paper. The gist The 1/ ln N acceleration for non-interacting systems is a statistical large-number effect: rare Brownian sprints become likely because many candidates are available.

Kai: Problem and Motivation Extreme first-passage events quantify when a noisy system reaches a prescribed set for the first time, providing fundamental timescales across physics, chemistry, biology, ecology, and information processing.

Mira: In many cases, the extreme first-passage time (EFPT) is more physically and biologically relevant because it asks for the earliest arrival among many searchers.

Kai: The central questions are therefore natural: are there any universal laws governing the extreme first-passage statistics of multiple searchers with generic interactions, and is it possible to exploit interactions to beat the fastest-search scale 1/ ln N of non-interacting searchers.

Title and authors: Kai: Let's talk about who wrote this. The paper is called "Extreme First-Passage Time of Many Interacting Particles," and the authors are Ruicheng Bao and others from the University of Tokyo, specifically from the Department of Physics.

Mira: It’s interesting to see a group tackling this complex many-body problem because it moves beyond just studying one particle or one searcher at a time.

Lev: For me, what catches my eye is that they are setting up this general framework for interacting extreme searches, which is the starting point for everything they build next.

Kai: They establish the initial setup with definitions like N i:= dist(X N i(zero),), which is just defining a target distance for each searcher i to the target set <ref:2607.22528#pg2,searcher $i$ to the target>.

Mira: And they define zero as the infimum of those distances, which basically sets the starting point for all these particles relative to the target <ref:2607.22528#pg2>.

Lev: What I need to stress is that this paper's main goal isn't just to prove something about a single interaction; it’s proving what universal statement remains true when you have many interacting particles.

Kai: The authors show that the natural question becomes what universal statement remains true with interaction because the factorization of the survival probability breaks down.

Mira: They are setting up the context for why we need these tools, which is that interactions are intrinsically many-body problems instead of just a product of one-searcher probabilities.

Lev: It’s setting the stage for proving whether you can exploit these interactions to beat that one/ N non-interacting scale, which is what they call the "natural question <ref:2607.22528#pg1>."

The paper's summary: Kai: So, what does the paper actually say about the results? They establish a no-go theorem showing that for broad classes of bounded interactions, you can't beat that one/ N extreme timescale for N independent Brownian searchers <ref:2607.22528#pg1,broad classes of bounded interactions>.

Mira: That means if your interaction forces are controlled and short-time drift is bounded by zero the probability of hitting the target in time t is upper-bounded by a term involving N erfc(zero - B N,t/sqrt 4Dt) <ref:2607.22528#pg2>.

Lev: That inequality then implies that the expected extreme first-passage time, ET N, has a lower bound of t times this probability as you go to the limit.

Kai: So you get an expected time that is greater than t times some expression involving the error function and those bounded forces. It shows that for these broad classes, the logarithmic scale is exact.

Mira: But they also establish a matching logarithmic upper bound when a large number of searchers, m N, start within an N-independent distance of a flat target.

Lev: And Proposition one shows that this upper bound is N to infinity (m N) ET N at most squared + 4D, and that confinement requires the total outward drift to be bounded by B N = o(m N) <ref:2607.22528#pg2>.

Kai: That matches up with Theorem one which gives the exact logarithmic universality class whenever both those hypotheses—the confinement and the drifting conditions—are met together <ref:2607.22528#pg1>.

The paper's improvements: Kai: Now, where does this get interesting for us? The authors identify two genuinely interaction-driven resources that you need to violate one of the ingredients of their no-go theorem to get faster scaling.

Mira: They say you have to violate either an order-one initial gap, or have sufficiently small deterministic transport on the logarithmic window, or change your ordinary Brownian fluctuations.

Lev: And they point out two specific mechanisms that can do this: first, screened single-file repulsion can create a coherent O(N) push on the leader.

Kai: If that happens, the expected time scales down to ET N at most lambda e L / (lambda g L - X N(zero)) N - one which is scaling like order N instead of logarithmic <ref:2607.22528#pg2>.

Mira: And the second mechanism is drift-free pairwise stochastic kicks, where every pair exchanges equal-and-opposite zero-mean kicks.

Lev: That stochastic forcing leads to a lower bound of N to infinity N N ET N = squared / (four kappa), which is faster than the logarithmic class <ref:2607.22528#pg2>.

Conclusion: Kai: So, to wrap up, the paper shows that a large class of interactions cannot create a new purely statistical acceleration of the leading scale and that general repulsion doesn't cause a leading-scale slowdown either.

Mira: They identify this sharp logarithmic class and specify exactly what physical conditions need to be met for it to hold true.

Lev: To get faster scaling, you need one of those deterministic large-number effects, like coherent targetward transport supported by the initial geometry or a change in the diffusive behavior itself.

Kai: The authors provide two explicit examples showing faster scaling and they derive a universal bound setting the fundamental limit for such interaction-driven acceleration.

Mira: It’s clear that interactions destroy factorization, but they also show where you can get away from the logarithmic class by engineering coherent drift or enhancing your fluctuations.

Lev: We'll keep an eye on these specific mechanisms because understanding how to engineer those forces is what will tell us if we can actually build systems that operate faster than the standard one/ N non-interacting limit <ref:2607.22528#pg1>.

Department of Physics, Graduate School of Science, The University of Tokyo

cond-mat.stat-mech, cond-mat.mes-hall, cond-mat.soft, math-ph, math.MP, physics.bio-ph

Submitted: 2026-07-24

Updated: 2026-10-20

Comments: 7+16 pages, submitted version. The case involving singular forces, including LJ and WCA interactions, has been resolved (a conjecture in v1 version). Additionally, a long-range repulsive example (Example C) has been added

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 90/100

The gist: The gist The 1/ ln N acceleration for non-interacting systems is a statistical large-number effect: rare Brownian sprints become likely because many candidates are available.

Key concepts

Extreme First-Passage Time (EFPT): This measures the time it takes for the very first particle among many to reach a specific target set. It is physically relevant because it represents the earliest arrival, which is often more important than the average time in complex systems like search or reaction kinetics.
No-Go Theorem for Bounded Interactions: This theorem proves that if interactions are controlled and short-ranged, they cannot fundamentally change the leading logarithmic scaling of the EFPT. It establishes a lower bound on arrival times that remains proportional to $1/ ext{ln } N$ unless specific conditions on drift or force bounds are violated.
Coherent Drift and Stochastic Kicks: These are two specific interaction-driven mechanisms that can overcome the logarithmic barrier. Coherent drift involves a synchronized push toward the target, while stochastic kicks involve pairwise, zero-mean random pushes between particles. These mechanisms allow the system to achieve faster scaling than non-interacting systems.
Logarithmic Universality Class: This is the standard scaling behavior for extreme first passage in this system when interactions are generic. The paper shows that most interactions fall into this class, meaning the time required scales as $1/ ext{ln } N$ unless a specific, non-generic interaction mechanism is present to accelerate it further.

Terminology

No-Go Theorem — 1× in this paper · explained in 2 episodes
Redundancy — 1× in this paper · explained in 2 episodes

Summary

The gist The 1/ ln N acceleration for non-interacting systems is a statistical large-number effect: rare Brownian sprints become likely because many candidates are available.

Problem and Motivation

Extreme first-passage events quantify when a noisy system reaches a prescribed set for the first time, providing fundamental timescales across physics, chemistry, biology, ecology, and information processing. In many cases, the extreme first-passage time (EFPT) is more physically and biologically relevant because it asks for the earliest arrival among many searchers. The central questions are therefore natural: are there any universal laws governing the extreme first-passage statistics of multiple searchers with generic interactions, and is it possible to exploit interactions to beat the fastest-search scale 1/ ln N of non-interacting searchers. Interactions destroy the factorization of the many-body survival probability into a product of one-searcher survival probabilities, making interacting extreme first passage intrinsically many-body.

No-Go Theorem for Bounded Interactions

The paper establishes a no-go theorem showing that interactions with controlled shorttime drift preserve the Brownian logarithmic barrier. Specifically, if the force supremum is bounded by BN,t and BN,tt < l0, then P(TN ≤ t) ≤ P(HcN,t) + N erfc l0 − BN,tt√4Dt, which implies E[TN] ≥ t 1 − P(HcN,t) − N erfc l0 − BN,tt√4Dt. This theorem permits the drift budget BN,t to grow with N; on the logarithmic window tN = c/ ln N, any BN,tN = O(ln N) still yields a 1/ ln N lower bound, whereas the sharp Brownian constant is recovered when BN,tN = o(ln N).

Logarithmic Upper Bound and Universality Class

The authors establish a matching logarithmic upper bound when a growing number mN → ∞ of searchers start within an Nindependent distance of a flat target. Proposition 1 states that the bound lim sup N→∞ (ln mN) E[TN] ≤ l 2+4D holds in either setting, where confinement requires the total outward drift to be bounded above by BN = o(ln mN). Without confinement, pure repulsion with targetward or zero one-body drift instead yields Eq. (10), from which the logarithmic mean bound follows. Combining Proposition 1 with Theorem 1 gives the exact logarithmic universality class whenever both hypotheses apply.

Mechanisms for Escaping the Logarithmic Class

To obtain a faster scaling, one must violate at least one of the ingredients of the no-go theorem: an order-one initial gap, sufficiently small deterministic transport on the logarithmic window, or ordinary Brownian-size fluctuations. Two genuinely interaction-driven resources remain after fixing the initial gap: coherent drift and enhancement of fluctuations.

Screened single-file repulsion can generate a coherent O(N) push on the leader, leading to E[TN] ≤ λeL/λg L − XN (0) N − 1 = O(N − 1).
Drift-free pairwise stochastic kicks where all pairs exchange equal-and-opposite zero-mean kicks leads to a lower bound of lim N→∞ N ln N ETN = l 20/4κ.

Fundamental Acceleration Limit

Theorem 2 provides a unified acceleration limit by interpolating between coherent drift and enhanced diffusion. The lower bound is given by t⋆N ≍ 1/ ln N when DeffN = D and BN = O(ln N), recovering the logarithmic scale of Theorem 1. For deterministic (p + 1)-body interactions whose contribution from each interacting (p + 1)-tuple is bounded, the force on any given searcher contains at most N−1 p = O(Np) interaction terms, leading to the fundamental N −p acceleration limit. For zero-mean (q + 1)-body stochastic interactions with bounded variance and orthogonal martingale increments, DeffN = O(Nq) and the lower-bound scale is (Nq ln N)−1.

Conclusion

The paper develops a general theory of extreme first passage for multiple interacting Brownian searchers. Interactions destroy the probability factorization, but Theorem 1 shows that a large class of interactions cannot create a new purely statistical acceleration of the leading scale and Proposition 1 shows that general repulsion does not create a leading-scale slowdown either. Together they identify a sharp logarithmic class and specify the physical conditions under which it holds. To obtain a faster scaling one needs a deterministic large-number effect, such as coherent targetward transport supported by the initial geometry, or a change in the diffusive behavior. The authors provide two explicit examples displaying faster scaling, and further derive a universal bound setting the fundamental limit of such interaction-driven acceleration.

End Matter

The convex projection used in Theorem 1 is not needed if one controls the full finite-variation displacement. For every η ∈ (0, 1) there is Kd,η < ∞ such that P(TN ≤ t) ≤ P(GN,t(a)) + NKd,η exp − (1 − η)2a 2/4Dmaxt. Consequently, if P(GN,c/ ln N (a)) → 0 for every c < a 2/(4Dmax), then lim infN→∞(ln N)ETN ≥ a 2/(4Dmax). For common diffusivity D, P(TN ≤ t) ≤ P((G∥N,t(a))c) + N erfc a√4Dt. Thus only targetward displacement must be controlled. For closed convex targets choose, at each initial point, the targetward supporting direction eN i used in Theorem 1. If G∥N,t(a):= max i sup s≤t∧TN eN i· A N i(s) ≤ l0 − a, then for common diffusivity D, P(TN ≤ t) ≤ P((G∥N,t(a))c) + N erfc a√4Dt.

B. Directional Relaxation for Convex Targets

For a closed convex target choose, at each initial point, the targetward supporting direction eN i used in Theorem 1. If G∥N,t(a):= max i sup s≤t∧TN eN i· A N i(s) ≤ l0 − a, then for common diffusivity D, P(TN ≤ t) ≤ P((G∥N,t(a))c) + N erfc a√4Dt.

C. Proof of Theorem 2

The first-hitter partition and the supporting-hyperplane inequality give P(TN ≤ t) ≤ εN (t) + NΠN (t,l0 − uN (t)). If uN (t) = BN t and ⟨MN i⟩s ≤ 2Deff N s on GN,t, Dambis–Dubins–Schwarz and the reflection principle yield ΠN (t, a) ≤ erfc(a/q4Deff N t) <ref:2607.

Improvements for AI systems

The AI system can achieve significantly faster decision-making in high-stakes, redundant signaling scenarios by exploiting coherent force accumulation as a mechanism to escape the logarithmic class. This allows for an improved efficiency scaling of N-p speed limits when deterministic (p + 1)-body forces are present.
The system can utilize stochastic coupling mechanisms to enhance search efficiency by implementing drift-free pairwise stochastic kicks, which provide an additional factor of (ln N) −1 in the lower bound scaling. This translates to a more efficient exploration strategy in environments where interactions are purely repulsive and drift is zero or targetward.
The AI can dynamically adapt its search strategy based on the initial configuration to maintain performance near the sharp Brownian constant when sublogarithmic local coordination is maintained, ensuring that the sharp Brownian constant l20/(4D) is recovered.