arXiv subjects

Quentin Berger

Publications and source records attributed to Quentin Berger.

At least 37 records · Page 2Linked to original sources

arXiv2021

The continuum directed polymer in Lévy Noise

We present in this paper the construction of a continuum directed polymer model in an environment given by space-time Lévy noise. One of the main objectives of this construction is to describe the scaling limit of discrete directed polymer in an heavy-tail environment and for this reason we put special emphasis on the case of $α$-stable noises with $α\in (1,2)$. Our construction can be performed in arbitrary dimension, provided that the Lévy measure satisfies specific (and dimension dependent) conditions. We also discuss a few basic properties of the continuum polymer and the relation between this model and the Stochastic Heat Equation with multiplicative Lévy noise.

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arXiv2021

The stochastic heat equation with multiplicative Lévy noise: Existence, moments, and intermittency

We study the stochastic heat equation (SHE) $\partial_t u = \frac12 Δu + βu ξ$ driven by a multiplicative Lévy noise $ξ$ with positive jumps and amplitude $β>0$, in arbitrary dimension $d\geq 1$. We prove the existence of solutions under an optimal condition if $d=1,2$ and a close-to-optimal condition if $d\geq3$. Under an assumption that is general enough to include stable noises, we further prove that the solution is unique. By establishing tight moment bounds on the multiple Lévy integrals arising in the chaos decomposition of $u$, we further show that the solution has finite $p$th moments for $p>0$ whenever the noise does. Finally, for any $p>0$, we derive upper and lower bounds on the moment Lyapunov exponents of order $p$ of the solution, which are asymptotically sharp in the limit as $β\to0$. One of our most striking findings is that the solution to the SHE exhibits a property called strong intermittency (which implies moment intermittency of all orders $p>1$ and pathwise mass concentration of the solution), for any non-trivial Lévy measure, at any disorder intensity $β>0$, in any dimension $d\geq1$.

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arXiv2021

The scaling limit of the directed polymer with power-law tail disorder

In this paper, we study the so-called intermediate disorder regime for a directed polymer in a random environment with heavy-tail. Consider a simple symmetric random walk $(S_n)_{n\geq 0}$ on $\mathbb{Z}^d$, with $d\geq 1$, and modify its law using Gibbs weights in the product form $\prod_{n=1}^{N} (1+βη_{n,S_n})$, where $(η_{n,x})_{n\ge 0, x\in \mathbb{Z}^d}$ is a field of i.i.d. random variables whose distribution satisfies $\mathbb{P}(η>z) \sim z^{-α}$ as $z\to\infty$, for some $α\in(0,2)$. We prove that if $α< \min(1+\frac{2}{d},2)$, when sending $N$ to infinity and rescaling the disorder intensity by taking $β=β_N \sim N^{-γ}$ with $γ=\frac{d}{2α}(1+\frac{2}{d}-α)$, the distribution of the trajectory under diffusive scaling converges in law towards a random limit, which is the continuum polymer with Lévy $α$-stable noise constructed in the companion paper arXiv:2007.06484.

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arXiv2019

Strong renewal theorems and local large deviations for multivariate random walks and renewals

We study a random walk $\mathbf{S}_n$ on $\mathbb{Z}^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbolα=(α_1,\ldots,α_d) \in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, $i.e.$ a sharp asymptotic of the Green function $G(\mathbf{0},\mathbf{x})$ as $\|\mathbf{x}\|\to +\infty$, along the "favorite direction or scaling": (i) if $\sum_{i=1}^d α_i^{-1} < 2$ (reminiscent of Garsia-Lamperti's condition when $d=1$ [Comm. Math. Helv. $\mathbf{37}$, 1962]); (ii) if a certain $local$ condition holds (reminiscent of Doney's condition [Probab. Theory Relat. Fields $\mathbf{107}$, 1997] when $d=1$). We also provide uniform bounds on the Green function $G(\mathbf{0},\mathbf{x})$, sharpening estimates when $\mathbf{x}$ is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case $α_i\equiv α$, in the favorite scaling, and has even left aside the case $α\in[1,2)$ with non-zero mean. Most of our estimates rely on new general (multivariate) local large deviations results, that were missing in the literature and that are of interest on their own.

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arXiv2019

Scaling Limit of Sub-ballistic 1D Random Walk among Biased Conductances: a Story of Wells and Walls

We consider a one-dimensional random walk among biased i.i.d. conductances, in the case where the random walk is transient but sub-ballistic: this occurs when the conductances have a heavy-tail at $+\infty$ or at $0$. We prove that the scaling limit of the process is the inverse of an $α$-stable subordinator, which indicates an aging phenomenon, expressed in terms of the generalized arcsine law. In analogy with the case of an i.i.d. random environment studied in details in [Enriquez, Sabot, Zindy, Bull. Soc. Math. 2009; Enriquez, Sabot, Tournier, Zindy, Ann. Appl. Probab. 2013], some `traps' are responsible for the slowdown of the random walk. However, the phenomenology is somehow different (and richer) here. In particular, three types of traps may occur, depending on the fine properties of the tails of the conductances: (i) a very large conductance (a well in the potential); (ii) a very small conductance (a wall in the potential); (iii) the combination of a large conductance followed shortly after by a small conductance (a well-and-wall in the potential).

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arXiv2018

Notes on Random Walks in the Cauchy Domain of Attraction

The goal of these notes is to fill some gaps in the literature about random walks in the Cauchy domain of attraction, which has been in many cases left aside because of its additional technical difficulties. We prove here several results in that case: a Fuk-Nagaev inequality and a local version of it ; a large deviation theorem ; two types of local large deviation theorems. We also derive two important applications of these results: a sharp estimate of the tail of the first ladder epochs, and renewal theorems -- extending standard renewal theorems to the case of random walks. Most of our techniques carry through to the case of random walks in the domain of attraction of an $α$-stable law with $α\in(0,2)$, so we also present results in that case, since some of them seem to be missing in the literature.

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arXiv2018

Disorder and denaturation transition in the generalized Poland-Scheraga model

We investigate the generalized Poland-Scheraga model, which is used in the bio-physical literature to model the DNA denaturation transition, in the case where the two strands are allowed to be non-complementary (and to have different lengths). The homogeneous model was recently studied from a mathematical point of view in Giacomin, Khatib (Stoch. Proc. Appl., 2017), via a $2$-dimensional renewal approach, with a loop exponent $2+α$ (${α>0}$): it was found to undergo a localization/delocalization phase transition of order $ν= \min(1,α)^{-1}$, together with -- in general -- other phase transitions. In this paper, we turn to the disordered model, and we address the question of the influence of disorder on the denaturation phase transition, that is whether adding an arbitrarily small amount of disorder (i.e. inhomogeneities) affects the critical properties of this transition. Our results are consistent with Harris' predictions for $d$-dimensional disordered systems (here $d=2$). First, we prove that when $α<1$ (i.e. $ν>d/2$), then disorder is irrelevant: the quenched and annealed critical points are equal, and the disordered denaturation phase transition is also of order $ν=α^{-1}$. On the other hand, when $α>1$, disorder is relevant: we prove that the quenched and annealed critical points differ. Moreover, we discuss a number of open problems, in particular the smoothing phenomenon that is expected to enter the game when disorder is relevant.

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arXiv2018

Geodesics Toward Corners in First Passage Percolation

For stationary first passage percolation in two dimensions, the existence and uniqueness of semi-infinite geodesics directed in particular directions or sectors has been considered by Damron and Hanson (Commun. Math. Phys., 2014), Ahlberg and Hoffman (preprint, 2016), and others. However the main results do not cover geodesics in the direction of corners of the limit shape $\mathcal{B}$, where two facets meet. We construct an example with the following properties: (i) the limiting shape is an octagon, (ii) semi-infinite geodesics exist only in the four axis directions, and (iii) in each axis direction there are multiple such geodesics. Consequently, the set of points of $\partial \mathcal{B}$ which are in the direction of some geodesic does not have all of $\mathcal{B}$ as its convex hull.

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arXiv2018

Directed polymers in heavy-tail random environment

We study the directed polymer model in dimension ${1+1}$ when the environment is heavy-tailed, with a decay exponent $α\in(0,2)$. We give all possible scaling limits of the model in the weak-coupling regime, i.e., when the inverse temperature temperature $β=β_n$ vanishes as the size of the system $n$ goes to infinity. When $α\in(1/2,2)$, we show that all possible transversal fluctuations $\sqrt{n} \leq h_n \leq n$ can be achieved by tuning properly $β_n$, allowing to interpolate between all super-diffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [cf:DZ] - we actually identify five different regimes. On the other hand, when $α<1/2$, we show that there are only two regimes: the transversal fluctuations are either $\sqrt{n}$ or $n$. As a key ingredient, we use the Entropy-controlled Last Passage Percolation (E-LPP), introduced in a companion paper [cf:BT_ELPP].

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arXiv2018

Beyond Hammersley's Last-Passage Percolation: a discussion on possible local and global constraints

Hammersley's Last-Passage Percolation (LPP), also known as Ulam's problem, is a well-studied model that can be described as follows: consider $m$ points chosen uniformly and independently in $[0,1]^2$, then what is the maximal number $\mathcal{L}_m$ of points that can be collected by an up-right path? We introduce here a generalization of this standard LPP, in order to allow for more general constraints than the up-right condition (a $1$-Lipschitz condition after rotation by $45^{\circ}$). We focus more specifically on two cases: (i) when the constraint is a $γ$-Hölder (local) condition, we call it H-LPP; (ii) when the constraint is a path-entropy (global) condition, we call it E-LPP. These generalizations also allows us to deal with non-directed LPP. We develop motivations for directed and non-directed constrained LPP, and we give the correct order of $\mathcal{L}_m$ in a general manner.

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arXiv2018

Entropy-controlled Last-Passage Percolation

In the present article we consider a natural generalization of Hammersley's Last Passage Percolation (LPP) called Entropy-controlled Last Passage Percolation (E-LPP), where points can be collected by paths with a global (entropy) constraint which takes in account the whole structure of the path, instead of a local ($1$-Lipschitz) constraint as in Hammersley's LPP. The E-LPP turns out to be a key ingredient in the context of the directed polymer model when the environment is heavy-tailed, which we consider in the related paper [Berger and Torri, 2018]. We prove several estimates on the E-LPP in continuous and in discrete settings, which are of interest on their own. We give applications in the context of polymers in heavy-tail environment which are essentials tools in [Berger and Torri, 2018]: we show that the limiting variational problem conjectured by [Dey and Zygouras, 2016] (Conjecture 1.7) is finite, and we prove that the discrete variational problem converges to the continuous one, generalizing techniques used by [Auffinger-Louidor, 2011] and [Hambly and Martin, 2007].

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arXiv2017

Disorder and critical phenomena: the $α=0$ copolymer model

The generalized copolymer model is a disordered system built on a discrete renewal process with inter-arrival distribution that decays in a regularly varying fashion with exponent $1+ α\geq 1$. It exhibits a localization transition which can be characterized in terms of the free energy of the model: the free energy is zero in the delocalized phase and it is positive in the localized phase. This transition, which is observed when tuning the mean $h$ of the disorder variable, has been tackled in the physics literature notably via a renormalization group procedure that goes under the name of \emph{strong disorder renormalization}. We focus on the case $α=0$ -- the critical value $h_c(β)$ of the parameter $h$ is exactly known (for every strength $β$ of the disorder) in this case -- and we provide precise estimates on the critical behavior. Our results confirm the strong disorder renormalization group prediction that the transition is of infinite order, namely that when $h\searrow h_c(β)$ the free energy vanishes faster than any power of $h-h_c(β)$. But we show that the free energy vanishes much faster than the physicists' prediction.

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arXiv2017

Scaling of sub-ballistic 1D Random Walks among biased Random Conductances

We consider two models of one-dimensional random walks among biased i.i.d. random conductances: the first is the classical exponential tilt of the conductances, while the second comes from the effect of adding an external field to a random walk on a point process (the bias depending on the distance between points). We study the case when the walk is transient to the right but sub-ballistic, and identify the correct scaling of the random walk: we find $α\in[0,1]$ such that $\log X_n / \log n \to α$. Interestingly, $α$ does not depend on the intensity of the bias in the first case, but it does in the second case.

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arXiv2017

Annealed scaling for a charged polymer in dimensions two and higher

This paper considers an undirected polymer chain on $\mathbb{Z}^d$, $d \geq 2$, with i.i.d.\ random charges attached to its constituent monomers. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. The object of interest is the \emph{annealed free energy} per monomer in the limit as the length $n$ of the polymer chain tends to infinity. We show that there is a critical curve in the parameter plane spanned by the charge bias and the inverse temperature separating an \emph{extended phase} from a \emph{collapsed phase}. We derive the scaling of the critical curve for small and for large charge bias and the scaling of the annealed free energy for small inverse temperature, which are both anomalous. We show that in a subset of the collapsed phase the polymer chain is \emph{subdiffusive}, namely, on scale $(n/\log n)^{1/(d+2)}$ it moves like a Brownian motion conditioned to stay inside a ball with a deterministic radius and a randomly shifted center. We expect this scaling to hold throughout the collapsed phase. We further expect that in the extended phase the polymer chain scales like a weakly self-avoiding walk. Proofs are based on a detailed analysis for simple random walk of the downward large deviations of the self-intersection local time and the upward large deviations of the range. Part of our scaling results are rough, and we formulate conjectures under which they can be sharpened. The existence of the free energy remains an open problem, which we are able to settle in a subset of the collapsed phase for a subclass of charge distributions.

math-ph↗

arXiv2017

Pinning of a renewal on a quenched renewal

We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process $σ$, and $0$ elsewhere, so nonzero potential values become sparse if the gaps in $σ$ have infinite mean. The "polymer" -- of length $σ_N$ -- is given by another renewal $τ$, whose law is modified by the Boltzmann weight $\exp(β\sum_{n=1}^N \mathbf{1}_{\{σ_n\inτ\}})$. Our assumption is that $τ$ and $σ$ have gap distributions with power-law-decay exponents $1+α$ and $1+\tilde α$ respectively, with $α\geq 0,\tilde α>0$. There is a localization phase transition: above a critical value $β_c$ the free energy is positive, meaning that $τ$ is \emph{pinned} on the quenched renewal $σ$. We consider the question of relevance of the disorder, that is to know when $β_c$ differs from its annealed counterpart $β_c^{\rm ann}$. We show that $β_c=β_c^{\rm ann}$ whenever $ α+\tilde α\geq 1$, and $β_c=0$ if and only if the renewal $τ\capσ$ is recurrent. On the other hand, we show $β_c>β_c^{\rm ann}$ when $ α+\frac32\, \tilde α<1$. We give evidence that this should in fact be true whenever $ α+\tilde α<1$, providing examples for all such $ α,\tilde α$ of distributions of $τ,σ$ for which $β_c>β_c^{\rm ann}$. We additionally consider two natural variants of the model: one in which the polymer and disorder are constrained to have equal numbers of renewals ($σ_N=τ_N$), and one in which the polymer length is $τ_N$ rather than $σ_N$. In both cases we show the critical point is the same as in the original model, at least when $ α>0$.

math.PR↗

arXiv2017

DNA melting structures in the generalized Poland-Scheraga model

The Poland-Scheraga model for DNA denaturation, besides playing a central role in applications, has been widely studied in the physical and mathematical literature over the past decades. More recently a natural generalization has been introduced in the biophysics literature to overcome the limits of the original model, namely to allow an excess of bases -- i.e. a different length of the two single stranded DNA chains -- and to allow slippages in the chain pairing. The increased complexity of the model is reflected in the appearance of configurational transitions when the DNA is in double stranded form. In a previous work of two of the authors the generalized Poland-Scheraga model has been analyzed thanks to a representation in terms of a bivariate renewal process. In this work we exploit this representation farther and fully characterize the path properties of the system, making therefore explicit the geometric structures -- and the configurational transitions -- that are observed when the polymer is in the double stranded form. What we prove is that, when the excess of bases is not absorbed in a homogeneous fashion along the double stranded chain, then it either condensates in a single macroscopic loop or it accumulates into an unbound single strand free end.

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arXiv2016

Local limit theorems and renewal theory with no moments

We study i.i.d. sums $τ_k$ of nonnegative variables with index $0$: this means $\mathbf{P}(τ_1=n) = φ(n) n^{-1}$, with $φ(\cdot)$ slowly varying, so that $\mathbf{E}(τ_1^\varepsilon)=\infty$ for all $\varepsilon>0$. We prove a local limit and local (upward) large deviation theorem, giving the asymptotics of $\mathbf{P}(τ_k=n)$ when $n$ is at least the typical length of $τ_k$. A recent renewal theorem by Nagaev [21] is an immediate consequence: $\mathbf{P}(n\inτ) \sim \mathbf{P}(τ_1=n)/\mathbf{P}(τ_1 > n)^2$ as $n\to\infty$. If instead we only assume regular variation of $\mathbf{P}(n\inτ)$ and slow variation of $U_n:= \sum_{k=0}^n \mathbf{P}(k\inτ)$, we obtain a similar equivalence but with $\mathbf{P}(τ_1=n)$ replaced by its average over a short interval. We give an application to the local asymptotics of the distribution of the first intersection of two independent renewals. We further derive downward moderate and large deviations estimates, that is, the asymptotics of $\mathbf{P}(τ_k \leq n)$ when $n$ is much smaller than the typical length of $τ_k$.

math.PR↗

arXiv2016

Local asymptotics for the first intersection of two independent renewals

We study the intersection of two independent renewal processes, $ρ=τ\capσ$. Assuming that $\mathbf{P}(τ_1 = n ) = φ(n)\, n^{-(1+α)}$ and $\mathbf{P}(σ_1 = n ) = \tildeφ(n)\, n^{-(1+ \tildeα)} $ for some $α,\tilde α\geq 0$ and some slowly varying $φ,\tildeφ$, we give the asymptotic behavior first of $\mathbf{P}(ρ_1>n)$ (which is straightforward except in the case of $\min(α,\tildeα)=1$) and then of $\mathbf{P}(ρ_1=n)$. The result may be viewed as a kind of reverse renewal theorem, as we determine probabilities $\mathbf{P}(ρ_1=n)$ while knowing asymptotically the renewal mass function $\mathbf{P}(n\inρ)=\mathbf{P}(n\inτ)\mathbf{P}(n\inσ)$. Our results can be used to bound coupling-related quantities, specifically the increments $|\mathbf{P}(n\inτ)-\mathbf{P}(n-1\inτ)|$ of the renewal mass function.

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