arXiv subjects

Nicolas Pétrélis

Publications and source records attributed to Nicolas Pétrélis.

At least 19 recordsLinked to original sources

arXiv2026

Weakly self-avoiding walk in a pareto-distributed random potential

We investigate a model of continuous-time simple random walk paths in $\mathbb{Z}^d$ undergoing two competing interactions: an attractive one towards the large values of a random potential, and a self-repellent one in the spirit of the well-known weakly self-avoiding random walk. We take the potential to be i.i.d.~Pareto-distributed with parameter $α>d$, and we tune the strength of the interactions in such a way that they both contribute on the same scale as $t\to\infty$. Our main results are (1) the identification of the logarithmic asymptotics of the partition function of the model in terms of a random variational formula, and, (2) the identification of the path behaviour that gives the overwhelming contribution to the partition function for $α>2d$: the random-walk path follows an optimal trajectory that visits each of a finite number of random lattice sites for a positive random fraction of time. We prove a law of large numbers for this behaviour, i.e., that all other path behaviours give strictly less contribution to the partition function. The joint distribution of the variational problem and of the optimal path can be expressed in terms of a limiting Poisson point process arising by a rescaling of the random potential. The latter convergence is in distribution and is in the spirit of a standard extreme-value setting for a rescaling of an i.i.d. potential in large boxes, like in \cite{KLMS09}.

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arXiv2025

Non-local random deposition models for earthquakes and energy propagation

We investigate a new class of non-local random deposition models, initially introduced by physicists to study the field of mechanical constraints (stress) applied along a line or on a given area located in a seismic zone. The non-local features are twofold. First, the falling objects have random and heavy-tailed dimensions. Second, the locations where the objects are falling are at least for some of the models that we consider, depending on the shape of the surface before deposition. We consider $(h_N)_{N\in \N}$ a sequence of random $(d+1)$-dimensional surfaces defined on $[0,D]^d$ for $d\in \{1,2\}$. Thus, the process $h_{N}$ is obtained by adding to $h_{N-1}$ an object $$\s\in [0,D]^d \mapsto Z_{N}^{α-1} ψ\Big(\frac{v_{Y_N}(\s)}{Z_N}\Big),$$ where $Z=(Z_i)_{i\in \N}$ is an i.i.d. sequence of Pareto random variables, where $ψ:[0,\infty)\mapsto \mathbb{R}^+$ determines the global shape of the object, where $v_{\y}(\x)$ is the distance between $\x$ and $\y$ on the torus and where $Y=(Y_i)_{i\in \N}$ are random variables in $[0,D]^d$ that provide the location of each falling object. In the present paper we focus on three variations of this model. First, the rand-model for which $Y$ is an i.i.d. sequence of uniform random variables. Then, the min-model, that introduces an important property of the physics of earthquakes but is also harder to tract since a strong correlation appears between the $N$-th falling object and the shape of the profile $h_{N-1}$. Finally we consider a variant of the rand-model: the stellar model, which allows us to study the intensity of the microwaves emitted by stellar clouds and measured at the Earth surface. For those three models, our results identify the limit in law of $(h_N)_{N\in \N}$ viewed as a sequence of continuous random functions rescaled properly. We also determine the limit in law of the fluctuations of $(h_N)_{N\in \N}$.

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arXiv2025

Phase diagram of the interacting partially directed self-avoiding walk attracted by a vertical wall

In the present paper, we consider the interacting partially-directed self-avoiding walk (IPDSAW) attracted by a vertical wall. The IPDSAW was introduced by Zwanzig and Lauritzen (J. Chem. Phys., 1968) as a manner of investigating the collapse transition of a homopolymer dipped in a repulsive solvent. We prove in particular that a surface transition occurs inside the collapsed phase between (i) a regime where the attractive vertical wall does not influence the geometry of the polymer and (ii) a regime where the polymer is partially attached at the wall on a length that is comparable to its horizontal extension, modifying its asymptotic Wulff shape. The latter rigorously confirms the conjecture exposed by physicists in (Physica A: Stat. Mech. \\& App., 2002). We push the analysis even further by providing sharp asymptotics of the partition function inside the collapsed phase.

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arXiv2021

A sharp asymptotics of the partition function for the collapsed interacting partially directed self-avoiding walk

In the present paper, we investigate the collapsed phase of the interacting partially-directed self-avoiding walk (IPDSAW) that was introduced in Zwanzig and Lauritzen (1968). We provide sharp asymptotics of the partition function inside the collapsed phase, proving rigorously a conjecture formulated in Guttmann (2015) and Owczarek et al. (1993). As a by-product of our result, we obtain that, inside the collapsed phase, a typical IPDSAW trajectory is made of a unique macroscopic bead, consisting of a concatenation of long vertical stretches of alternating signs, outside which only finitely many monomers are lying.

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arXiv2020

Surface transition in the collapsed phase of a self-interacting walk adsorbed along a hard wall

The present paper is dedicated to the 2-dimensional Interacting Partially Directed Self Avoiding Walk constrained to remain in the upper-half plan and interacting with the horizontal axis. The model has been introduced in \cite{F90} to investigate the behavior of a homopolymer dipped in a poor solvent and adsorbed along a horizontal hard wall. It is known to undergo a collapse transition between an extended phase, inside which typical configurations of the polymer have a large horizontal extension (comparable to their total size), and a collapsed phase inside which the polymer looks like a globule. In the present paper, we establish rigorously that inside the collapsed phase, a surface transition occurs between an adsorbed-collapsed regime where the bottommost layer of the globule is pinned at the hard wall, and a desorbed-collapsed regime where the globule wanders away from the wall. To prove the existence of this surface transition and exhibit its associated critical curve, we display some sharp asymptotics of the partition function for a slightly simplified version of the model.

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arXiv2019

Limit theorems for Random Walk excursion conditioned to have a typical area

We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to zero and a finite fourth moment. This result complements the work of \citep{DKW13} where local central limit theorems are provided for the geometric area of the excursion of a symmetric random walk with finite second moments. Our result turns out to be a key tool to derive the scaling limit of the \emph{Interacting Partially-Directed Self-Avoiding Walk} at criticality which is the object of a companion paper \citep{CarPet17a}. This requires to derive a reinforced version of our result in the case of a random walk with Laplace symmetric increments.

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arXiv2018

Interacting partially directed self-avoiding walk: a probabilistic perspective

We review some recent results obtained in the framework of the 2-dimensional Interacting Self-Avoiding Walk (ISAW). After a brief presentation of the rigorous results that have been obtained so far for ISAW we focus on the Interacting Partially Directed Self-Avoiding Walk (IPDSAW), a model introduced in Zwanzig and Lauritzen (1968) to decrease the mathematical complexity of ISAW. In the first part of the paper, we discuss how a new probabilistic approach based on a random walk representation (see Nguyen and Pétrélis (2013)) allowed for a sharp determination of the asymptotics of the free energy close to criticality (see Carmona, Nguyen and Pétrélis (2016)). Some scaling limits of IPDSAW were conjectured in the physics literature (see e.g. Brak et al. (1993)). We discuss here the fact that all limits are now proven rigorously, i.e., for the extended regime in Carmona and Pétrélis (2016), for the collapsed regime in Carmona, Nguyen and Pétrélis (2016) and at criticality in Carmona and Pétrélis (2017a). The second part of the paper starts with the description of four open questions related to physically relevant extensions of IPDSAW. Among such extensions is the Interacting Prudent Self-Avoiding Walk (IPSAW) whose configurations are those of the 2-dimensional prudent walk. We discuss the main results obtained in Pétrélis and Torri (2016+) about IPSAW and in particular the fact that its collapse transition is proven to exist rigorously.

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arXiv2017

A shape theorem for the scaling limit of the IPDSAW at criticality

In this paper we give a complete characterization of the scaling limit of the critical Interacting Partially Directed Self-Avoiding Walk (IPDSAW) introduced in Zwanzig and Lauritzen (1968). As the system size $L$ diverges, we prove that the set of occupied sites, rescaled horizontally by $L^{2/3}$ and vertically by $L^{1/3}$ converges in law for the Hausdorff distance towards a non trivial random set. This limiting set is built with a Brownian motion $B$ conditioned to come back at the origin at $a_1$ the time at which its geometric area reaches $1$. The modulus of $B$ up to $a_1$ gives the height of the limiting set, while its center of mass process is an independent Brownian motion. Obtaining the shape theorem requires to derive a functional central limit theorem for the excursion of a random walk with Laplace symmetric increments conditioned on sweeping a prescribed geometric area. This result is proven in a companion paper arXiv:1709.06448.

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arXiv2017

Scaling limit of the uniform prudent walk

We study the 2-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the direction of a previously visited site. The uniform prudent walk has been investigated with combinatorial techniques in [Bousquet-Mélou, 2010], while another variant, the kinetic prudent walk has been analyzed in detail in [Beffara, Friedli and Velenik, 2010]. In this paper, we prove that the $2$-dimensional uniform prudent walk is ballistic and follows one of the $4$ diagonals with equal probability. We also establish a functional central limit theorem for the fluctuations of the path around the diagonal.

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arXiv2016

Collapse transition of the interacting prudent walk

This article is dedicated to the study of the 2-dimensional interacting prudent self-avoiding walk (referred to by the acronym IPSAW) and in particular to its collapse transition. The interaction intensity is denoted by $β>0$ and the set of trajectories consists of those self-avoiding paths respecting the prudent condition, which means that they do not take a step towards a previously visited lattice site. The IPSAW interpolates between the interacting partially directed self-avoiding walk (IPDSAW) that was analyzed in details in, e.g., [Zwanzig and Lauritzen, 1968], [Brak et al., 1992], [Carmona, Nguyen and Pétrélis, 2013-2016], and the interacting self-avoiding walk (ISAW) for which the collapse transition was conjectured in [Saleur, 1986]. Three main theorems are proven. We show first that IPSAW undergoes a collapse transition at finite temperature and, up to our knowledge, there was so far no proof in the literature of the existence of a collapse transition for a non-directed model built with self-avoiding path. We also prove that the free energy of IPSAW is equal to that of a restricted version of IPSAW, i.e., the interacting two-sided prudent walk. Such free energy is computed by considering only those prudent path with a general north-east orientation. As a by-product of this result we obtain that the exponential growth rate of generic prudent paths equals that of two-sided prudent paths and this answers an open problem raised in e.g., [Bousquet-Mélou, 2010] or [Dethridge and Guttmann, 2008]. Finally we show that, for every $β>0$, the free energy of ISAW itself is always larger than $β$ and this rules out a possible self-touching saturation of ISAW in its conjectured collapsed phase.

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arXiv2016

Phase diagram for a copolymer in a micro-emulsion

In this paper we study a model describing a copolymer in a micro-emulsion. The copolymer consists of a random concatenation of hydrophobic and hydrophilic monomers, the micro-emulsion consists of large blocks of oil and water arranged in a percolation-type fashion. The interaction Hamiltonian assigns energy $-α$ to hydrophobic monomers in oil and energy $-β$ to hydrophilic monomers in water, where $α,β$ are parameters that without loss of generality are taken to lie in the cone $\{(α,β) \in\mathbb{R}^2\colon\,α\geq |β|\}$. Depending on the values of these parameters, the copolymer either stays close to the oil-water interface (localization) or wanders off into the oil and/or the water (delocalization). Based on an assumption about the strict concavity of the free energy of a copolymer near a linear interface, we derive a variational formula for the quenched free energy per monomer that is column-based, i.e., captures what the copolymer does in columns of different type. We subsequently transform this into a variational formula that is slope-based, i.e., captures what the polymer does as it travels at different slopes, and we use the latter to identify the phase diagram in the $(α,β)$-cone. There are two regimes: supercritical (the oil blocks percolate) and subcritical (the oil blocks do not percolate). The supercritical and the subcritical phase diagram each have two localized phases and two delocalized phases, separated by four critical curves meeting at a quadruple critical point. The different phases correspond to the different ways in which the copolymer can move through the micro-emulsion. The analysis of the phase diagram is based on three hypotheses of percolation-type on the blocks. We show that these three hypotheses are plausible, but do not provide a proof.

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arXiv2015

Interacting partially directed self avoiding walk : scaling limits

This paper is dedicated to the investigation of a $1+1$ dimensional self-interacting and partially directed self-avoiding walk, usually referred to by the acronym IPDSAW and introduced in \cite{ZL68} by Zwanzig and Lauritzen to study the collapse transition of an homopolymer dipped in a poor solvant. In \cite{POBG93}, physicists displayed numerical results concerning the typical growth rate of some geometric features of the path as its length $L$ diverges. From this perspective the quantities of interest are the projections of the path onto the horizontal axis (also called horizontal extension) and onto the vertical axis for which it is useful to define the lower and the upper envelopes of the path. With the help of a new random walk representation, we proved in \cite{CNGP13} that the path grows horizontally like $\sqrt{L}$ in its collapsed regime and that, once rescaled by $\sqrt{L}$ vertically and horizontally, its upper and lower envelopes converge to some deterministic Wulff shapes. In the present paper, we bring the geometric investigation of the path several steps further. In the extended regime, we prove a law of large number for the horizontal extension of the polymer rescaled by its total length $L$, we provide a precise asymptotics of the partition function and we show that its lower and upper envelopes, once rescaled in time by $L$ and in space by $\sqrt{L}$, converge to the same Brownian motion. At criticality, we identify the limiting distribution of the horizontal extension rescaled by $L^{2/3}$ and we show that the excess partition function decays as $L^{2/3}$ with an explicit prefactor. In the collapsed regime, we identify the joint limiting distribution of the fluctuations of the upper and lower envelopes around their associated limiting Wulff shapes, rescaled in time by $\sqrt{L}$ and in space by $L^{1/4}$.

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arXiv2012

Lectures on random polymers

These lecture notes are a guided tour through the fascinating world of polymer chains interacting with themselves and/or with their environment. The focus is on the mathematical description of a number of physical and chemical phenomena, with particular emphasis on phase transitions and space-time scaling. The topics covered, though only a selection, are typical for the area. Sections 1-3 describe models of polymers without disorder, Sections 4-6 models of polymers with disorder. Appendices A-E contain tutorials in which a number of key techniques are explained in more detail.

math.PR↗

arXiv2012

Free energy of a copolymer in a micro-emulsion

In this paper we consider a two-dimensional model of a copolymer consisting of a random concatenation of hydrophilic and hydrophobic monomers, immersed in a micro-emulsion of random droplets of oil and water. The copolymer interacts with the micro-emulsion through an interaction Hamiltonian that favors matches and disfavors mismatches between the monomers and the solvents, in such a way that the interaction with the oil is stronger than with the water. The configurations of the copolymers are directed self-avoiding paths in which only steps up, down and right are allowed. The configurations of the micro-emulsion are square blocks with oil and water arranged in percolation-type fashion. The only restriction imposed on the path is that in every column of blocks its vertical displacement on the block scale is bounded. The way in which the copolymer enters and exits successive columns of blocks is a directed self-avoiding path as well, but on the block scale. We refer to this path as the coarse-grained self-avoiding path. We are interested in the limit as the copolymer and the blocks become large, in such a way that the copolymer spends a long time in each block yet visits many blocks. This is a coarse-graining limit in which the space-time scales of the copolymer and of the micro-emulsion become separated. We derive a variational formula for the quenched free energy per monomer, where quenched means that the disorder in the copolymer and the disorder in the micro-emulsion are both frozen. In a sequel paper we will analyze this variational formula and identify the phase diagram. It turns out that there are two regimes, supercritical and subcritical, depending on whether the oil blocks percolate or not along the coarse-grained self-avoiding path. The phase diagrams in the two regimes turn out to be completely different.

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arXiv2010

The discrete-time parabolic Anderson model with heavy-tailed potential

We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed (1+d)-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the d orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically. We give an explicit characterization of the localization point and of the typical paths of the model.

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arXiv2009

A polymer in a multi-interface medium

We consider a model for a polymer chain interacting with a sequence of equispaced flat interfaces through a pinning potential. The intensity $δ\in \mathbb {R}$ of the pinning interaction is constant, while the interface spacing $T=T_N$ is allowed to vary with the size $N$ of the polymer. Our main result is the explicit determination of the scaling behavior of the model in the large $N$ limit, as a function of $(T_N)_N$ and for fixed $δ>0$. In particular, we show that a transition occurs at $T_N=O(\log N)$. Our approach is based on renewal theory.

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arXiv2009

Depinning of a polymer in a multi-interface medium

In this paper we consider a model which describes a polymer chain interacting with an infinity of equi-spaced linear interfaces. The distance between two consecutive interfaces is denoted by T = T_N and is allowed to grow with the size N of the polymer. When the polymer receives a positive reward for touching the interfaces, its asymptotic behavior has been derived in a previous paper, showing that a transition occurs when T_N \approx log(N). In the present paper, we deal with the so-called depinning case, i.e., the polymer is repelled rather than attracted by the interfaces. Using techniques from renewal theory, we determine the scaling behavior of the model for large N as a function of T_N, showing that two transitions occur, when T_N \approx N^{1/3} and when T_N \approx N^{1/2} respectively.

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arXiv2007

On the localized phase of a copolymer in an emulsion: supercritical percolation regime

In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, $A$ and $B$, each occurring with density 1/2. The emulsion is a random mixture of liquids of two types, $A$ and $B$, organised in large square blocks occurring with density $p$ and $1-p$, respectively, where $p \in (0,1)$. The copolymer in the emulsion has an energy that is minus $α$ times the number of $AA$-matches minus $β$ times the number of $BB$-matches, where without loss of generality the interaction parameters can be taken from the cone $\{(α,β)\in\R^2\colon α\geq |β|\}$. To make the model mathematically tractable, we assume that the copolymer is directed and can only enter and exit a pair of neighbouring blocks at diagonally opposite corners. In \cite{dHW06}, it was found that in the supercritical percolation regime $p \geq p_c$, with $p_c$ the critical probability for directed bond percolation on the square lattice, the free energy has a phase transition along a curve in the cone that is independent of $p$. At this critical curve, there is a transition from a phase where the copolymer is fully delocalized into the $A$-blocks to a phase where it is partially localized near the $AB$-interface. In the present paper we prove three theorems that complete the analysis of the phase diagram : (1) the critical curve is strictly increasing; (2) the phase transition is second order; (3) the free energy is infinitely differentiable throughout the partially localized phase.

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