arXiv subjects

Léo Régnier

Publications and source records attributed to Léo Régnier.

9 recordsLinked to original sources

arXiv2026

Evolutionary Advantage of Diversity-Generating Retroelements in Switching Environments

Diversity-Generating Retroelements (DGRs) create rapid, targeted variation within specific genomic regions in phages and bacteria. They operate through stochastic retro-transcription of a template region (TR) into a variable region (VR), which typically encodes ligand-binding proteins. Despite their prevalence, the evolutionary conditions that maintain such hypermutating systems remain unclear. Here we introduce a two-timescale framework separating fast VR diversification from slow TR evolution, allowing the dynamics of DGR-controlled loci to be analytically understood from the TR design point of view. We quantity the fitness gain provided by the diversification mechanism of DGR in the presence of environmental switching with respect to standard mutagenesis. Our framework accounts for observed patterns of DGR activity in human-gut \textit{Bacteroides} and clarifies when constitutive DGR activation is evolutionarily favored.

q-bio.PE↗

arXiv2025

Flips Reveal the Universal Impact of Memory on Random Explorations

Quantifying space exploration is a central question in random walk theory, with direct applications ranging from animal foraging, diffusion-limited reactions, and intracellular transport to stock markets. In particular, the explored domain by one or many simple memoryless (or Markovian) random walks has received considerable attention . However, the physical systems mentioned above typically involve significant memory effects, and so far, no general framework exists to analyze such systems. We introduce the concept of a \emph{flip}, defined most naturally in one dimension, where the visited territory is $[x_{\rm min}, x_{\rm max}]$: a flip occurs when, after discovering a new site at $x_{\rm max}$, the walker next discovers $x_{\rm min} - 1$ instead of $x_{\rm max} + 1$ (and vice-versa). While it reduces to the classical splitting probability in Markovian systems, we show that the flip probability serves as a key observable for quantifying the impact of memory effects on space exploration. Here, we demonstrate that the flip probability follows a strikingly simple and universal law: it decays inversely with the number of sites visited, as $1/n$, independently of the underlying stochastic process. We confirm this behavior through simulations across paradigmatic non-Markovian models and observe it in real-world systems, without relying on model assumptions, including biological tracer motion, DNA sequences, and financial market dynamics. Finally, we reveal the physical mechanism behind this universality and show how it extends to higher-dimensional and fractal domains. Our determination of universal flip statistics lay the groundwork for understanding how memory effects govern random explorations.

cond-mat.stat-mech↗

arXiv2024

Persistence exponents of self-interacting random walks

The persistence exponent, which characterises the long-time decay of the survival probability of stochastic processes in the presence of an absorbing target, plays a key role in quantifying the dynamics of fluctuating systems. Determining this exponent for non-Markovian processes is known to be a difficult task, and exact results remain scarce despite sustained efforts. In this Letter, we consider the fundamental class of self-interacting random walks (SIRWs), which display long-range memory effects that result from the interaction of the random walker at time $t$ with the territory already visited at earlier times $t'<t$. We compute exactly the persistence exponent for all physically relevant SIRWs. As a byproduct, we also determine the splitting probability of these processes. Besides their intrinsic theoretical interest, these results provide a quantitative characterization of the exploration process of SIRWs, which are involved in fields as diverse as foraging theory, cell biology, and machine learning.

cond-mat.stat-mech↗

arXiv2024

Starving Random Walks

In this chapter, we review recent results on the starving random walk (RW) problem, a minimal model for resource-limited exploration. Initially, each lattice site contains a single food unit, which is consumed upon visitation by the RW. The RW starves whenever it has not found any food unit within the previous $\mathcal{S}$ steps. To address this problem, the key observable corresponds to the inter-visit time $τ_k$ defined as the time elapsed between the finding of the $k^\text{th}$ and the $(k+1)^\text{th}$ food unit. By characterizing the maximum $M_n$ of the inter-visit times $τ_0,\dots,τ_{n-1}$, we will see how to obtain the number $N_\mathcal{S}$ of food units collected at starvation, as well as the lifetime $T_\mathcal{S}$ of the starving RW.

cond-mat.stat-mech↗

arXiv2024

Full Record Statistics of 1d Random Walks

We develop a comprehensive framework for analyzing full record statistics, covering record counts $M(t_1), M(t_2), \ldots$, and their corresponding attainment times $T_{M(t_1)}, T_{M(t_2)}, \ldots$, as well as the intervals until the next record. From this multiple-time distribution, we derive general expressions for various observables related to record dynamics, including the conditional number of records given the number observed at a previous time and the conditional time required to reach the current record, given the occurrence time of the previous one. Our formalism is exemplified by a variety of stochastic processes, including biased nearest-neighbor random walks, asymmetric run-and-tumble dynamics, and random walks with stochastic resetting.

cond-mat.stat-mech↗

arXiv2023

Record Ages of Scale Invariant non-Markovian Random Walks

How long is needed for an observable to exceed its previous highest value and establish a new record? This time, known as the age of a record plays a crucial role in quantifying record statistics. Until now, general methods for determining record age statistics have been limited to observations of either independent random variables or successive positions of a Markovian (memoryless) random walk. Here we develop a theoretical framework to determine record age statistics in the presence of memory effects for continuous non-smooth processes that are asymptotically scale-invariant. Our theoretical predictions are confirmed by numerical simulations and experimental realizations of diverse representative non-Markovian random walk models and real time series with memory effects, in fields as diverse as genomics, climatology, hydrology, geology and computer science. Our results reveal the crucial role of the number of records already achieved in time series and change our view on analysing record statistics.

cond-mat.stat-mech↗

arXiv2023

Universal exploration dynamics of random walks

The territory explored by a random walk is a key property that may be quantified by the number of distinct sites that the random walk visits up to a given time. The extent of this spatial exploration characterizes many important physical, chemical, and ecological phenomena. In spite of its fundamental interest and wide utility, the number of visited sites gives only an incomplete picture of this exploration. In this work, we introduce a more fundamental quantity, the elapsed time $τ_n$ between visits to the $n^{\rm th}$ and the $(n+1)^{\rm st}$ distinct sites, from which the full dynamics about the visitation statistics can be obtained. To determine the distribution of these inter-visit times $τ_n$, we develop a theoretical approach that relies on a mapping with a trapping problem, in which, in contrast to previously studied situations, the spatial distribution of traps is continuously updated by the random walk itself. Despite the geometrical complexity of the territory explored by a random walk (typically aspherical, as well as containing holes and islands at all scales), we find that the distribution of the $τ_n$ can be accounted for by simple analytical expressions. Processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes for the temporal history of distinct sites visited. We confirm our theoretical predictions by Monte Carlo and exact enumeration methods. We also determine additional basic exploration observables, such as the perimeter of the visited domain or the number of islands of unvisited sites enclosed within this domain, thereby illustrating the generality of our approach. Because of their fundamental character and their universality, these inter-visit times represent a promising tool to unravel many more aspects of the exploration dynamics of random walks.

cond-mat.stat-mech↗

arXiv2022

Complete Visitation Statistics of 1d Random Walks

We develop a framework to determine the complete statistical behavior of a fundamental quantity in the theory of random walks, namely, the probability that $n_1$, $n_2$, $n_3$, . . . distinct sites are visited at times $t_1$, $t_2$, $t_3$, ... . From this multiple-time distribution, we show that the visitation statistics of 1d random walks are temporally correlated and we quantify the non-Markovian nature of the process. We exploit these ideas to derive unexpected results for the two-time trapping problem and also to determine the visitation statistics of two important stochastic processes, the run-and-tumble particle and the biased random walk.

cond-mat.stat-mech↗

arXiv2022

From individual-based epidemic models to McKendrick-von Foerster PDEs: A guide to modeling and inferring COVID-19 dynamics

We present a unifying, tractable approach for studying the spread of viruses causing complex diseases requiring to be modeled using a large number of types (e.g., infective stage, clinical state, risk factor class). We show that recording each infected individual's infection age, i.e., the time elapsed since infection, has three benefits. First, regardless of the number of types, the age distribution of the population can be described by means of a first-order, one-dimensional partial differential equation (PDE) known as the McKendrick-von Foerster equation. The frequency of type $i$ is simply obtained by integrating the probability of being in state $i$ at a given age against the age distribution. This representation induces a simple methodology based on the additional assumption of Poisson sampling to infer and forecast the epidemic. We illustrate this technique using French data from the COVID-19 epidemic. Second, our approach generalizes and simplifies standard compartmental models using high-dimensional systems of ordinary differential equations (ODEs) to account for disease complexity. We show that such models can always be rewritten in our framework, thus, providing a low-dimensional yet equivalent representation of these complex models. Third, beyond the simplicity of the approach, we show that our population model naturally appears as a universal scaling limit of a large class of fully stochastic individual-based epidemic models, where the initial condition of the PDE emerges as the limiting age structure of an exponentially growing population starting from a single individual.

q-bio.PE↗