Quentin Berger
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View article: The Random Walk Pinning Model I: Lower bounds on the free energy and disorder irrelevance
The Random Walk Pinning Model I: Lower bounds on the free energy and disorder irrelevance Open
The Random Walk Pinning Model (RWPM) is a statistical mechanics model in which the trajectory of a continuous time random walk $X=(X_t)_{t\geq 0}$ is rewarded according to the time it spends together with a moving catalyst. More specifical…
View article: The Random Walk Pinning Model II: Upper bounds on the free energy and disorder relevance
The Random Walk Pinning Model II: Upper bounds on the free energy and disorder relevance Open
This article investigates the question of disorder relevance for the continuous-time Random Walk Pinning Model (RWPM) and completes the results of our companion paper. The RWPM considers a continuous time random walk $X=(X_t)_{t\geq 0}$, w…
View article: Some properties of the principal Dirichlet eigenfunction in Lipschitz domains, via probabilistic couplings
Some properties of the principal Dirichlet eigenfunction in Lipschitz domains, via probabilistic couplings Open
We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set Ω ⊂ R d , in dimension d ≥ 2. More precisely, consider the simple random walk on Z d killed upon exiting the (large) bounded doma…
View article: Strong disorder for Stochastic Heat Flow and 2D Directed Polymers
Strong disorder for Stochastic Heat Flow and 2D Directed Polymers Open
The critical 2D Stochastic Heat Flow (SHF) is a universal measure-valued process providing a notion of solution to the ill-defined 2D stochastic heat equation. We investigate the SHF in the regime of large time and large disorder strength,…
View article: Collective vs. individual behaviour for sums of i.i.d. random variables: appearance of the one-big-jump phenomenon
Collective vs. individual behaviour for sums of i.i.d. random variables: appearance of the one-big-jump phenomenon Open
This article studies large and local large deviations for sums of i.i.d. real-valued random variables in the domain of attraction of an -stable law, , with emphasis on the case . There are two different scenarios: either the deviation is r…
View article: A random polymer approach to the weak disorder phase of the vertex reinforced jump process
A random polymer approach to the weak disorder phase of the vertex reinforced jump process Open
In this paper, we study the transient phase of the Vertex Reinforced Jump Process (VRJP) in dimension $d\geq 3$. In Sabot, Zeng (2019), the authors introduce a positive martingale and show that the VRJP is recurrent if and only if that mar…
View article: Some topics in random walks
Some topics in random walks Open
We collect a few recent results on random walks, which are ubiquitous in probability theory. The topics covered are: persistence problems for stochastic processes, large fluctuations in multi-scale modeling for rest hematopoiesis, and fine…
View article: Some properties of the principal Dirichlet eigenfunction in Lipschitz domains, via probabilistic couplings
Some properties of the principal Dirichlet eigenfunction in Lipschitz domains, via probabilistic couplings Open
We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set $Ω\subset \mathbb{R}^d$, in dimension $d\geq 2$. More precisely, consider the simple random walk on $\mathbb{Z}^d$ killed upon ex…
View article: On joint returns to zero of Bessel processes
On joint returns to zero of Bessel processes Open
In this article, we consider joint returns to zero of $n$ Bessel processes ($n\geq 2$): our main goal is to estimate the probability that they avoid having joint returns to zero for a long time. More precisely, considering $n$ independent …
View article: Non-linear conductances of Galton-Watson trees and application to the (near) critical random cluster model
Non-linear conductances of Galton-Watson trees and application to the (near) critical random cluster model Open
In this article, we study concave recursions on trees, which appear widely in information theory through algorithms such as belief propagation, and in statistical mechanics through models on tree-like graphs, including the Ising model, per…
View article: Ising model on a Galton–Watson tree with a sparse random external field
Ising model on a Galton–Watson tree with a sparse random external field Open
54 pages, comments welcome
View article: Ising model on a Galton-Watson tree with a sparse random external field
Ising model on a Galton-Watson tree with a sparse random external field Open
We consider the Ising model on a supercritical Galton-Watson tree $\mathbf{T}_n$ of depth $n$ with a sparse random external field, given by a collection of i.i.d. Bernouilli random variables with vanishing parameter $p_n$. This may me view…
View article: An application of Sparre Andersen's fluctuation theorem for exchangeable and sign-invariant random variables
An application of Sparre Andersen's fluctuation theorem for exchangeable and sign-invariant random variables Open
We revisit here a famous result by Sparre Andersen on persistence probabilities $\mathbf{P}(S_k>0 \;\forall\, 0\leq k\leq n)$ for symmetric random walks $(S_n)_{n\geq 0}$. We give a short proof of this result when considering sums of rando…
View article: Persistence problems for additive functionals of one-dimensional Markov processes
Persistence problems for additive functionals of one-dimensional Markov processes Open
In this article, we consider additive functionals $ζ_t = \int_0^t f(X_s)\mathrm{d} s$ of a càdlàg Markov process $(X_t)_{t\geq 0}$ on $\mathbb{R}$. Under some general conditions on the process $(X_t)_{t\geq 0}$ and on the function $f$, we …
View article: Collective vs. individual behaviour for sums of i.i.d. random variables: appearance of the one-big-jump phenomenon
Collective vs. individual behaviour for sums of i.i.d. random variables: appearance of the one-big-jump phenomenon Open
This article studies large and local large deviations for sums of i.i.d. real-valued random variables in the domain of attraction of an $α$-stable law, $α\in (0,2]$, with emphasis on the case $α=2$. There are two different scenarios: eithe…
View article: Scaling limit of the disordered generalized Poland--Scheraga model for DNA denaturation
Scaling limit of the disordered generalized Poland--Scheraga model for DNA denaturation Open
The Poland--Scheraga model, introduced in the 1970's, is a reference model to describe the denaturation transition of DNA. More recently, it has been generalized in order to allow for asymmetry in the strands lengths and in the formation o…
View article: The continuum directed polymer in Lévy noise
The continuum directed polymer in Lévy noise Open
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 a discrete directed polym…
View article: Non-directed polymers in heavy-tail random environment in dimension d≥2
Non-directed polymers in heavy-tail random environment in dimension d≥2 Open
In this article we study a non-directed polymer model in dimension d≥2: we consider a simple symmetric random walk on Zd which interacts with a random environment, represented by i.i.d. random variables (ωx)x∈Zd. The model consists in modi…
View article: One-dimensional polymers in random environments: stretching vs. folding
One-dimensional polymers in random environments: stretching vs. folding Open
In this article we study a \\emph{non-directed polymer model} on $\\mathbb Z$,\nthat is a one-dimensional simple random walk placed in a random environment.\nMore precisely, the law of the random walk is modified by the exponential of\nthe…
View article: Beyond Hammersley’s Last-Passage Percolation: a discussion on possible local and global constraints
Beyond Hammersley’s Last-Passage Percolation: a discussion on possible local and global constraints Open
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 $\mat…
View article: Non-directed polymers in heavy-tail random environment in dimension $d\geq 2$
Non-directed polymers in heavy-tail random environment in dimension $d\geq 2$ Open
In this article we study a \emph{non-directed} polymer model in dimension $d\ge 2$: we consider a simple symmetric random walk on $\mathbb{Z}^d$ which interacts with a random environment, represented by i.i.d. random variables $(ω_x)_{x\in…
View article: Mobility as a Service: An Exploratory Study of Consumer Mobility Behaviour
Mobility as a Service: An Exploratory Study of Consumer Mobility Behaviour Open
Key challenges in transportation need to be addressed to tackle the problems of fossil fuel emissions and worsened air quality in urban area. The development of a more efficient and clean transport system could benefit from mobility as a s…
View article: Disorder and denaturation transition in the generalized Poland–Scheraga model
Disorder and denaturation transition in the generalized Poland–Scheraga model Open
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 lengt…
View article: Scaling limit of sub-ballistic 1D random walk among biased conductances: a story of wells and walls
Scaling limit of sub-ballistic 1D random walk among biased conductances: a story of wells and walls Open
We consider a one-dimensional random walk among biased i.i.d. conductances,\nin the case where the random walk is transient but sub-ballistic: this occurs\nwhen the conductances have a heavy-tail at $+\\infty$ or at $0$. We prove that\nthe…
View article: Directed polymers in heavy-tail random environment
Directed polymers in heavy-tail random environment Open
We study the directed polymer model in dimension ${1+1}$ when the environment is heavy-tailed, with a decay exponent $\\alpha \\in (0,2)$. We give all possible scaling limits of the model in the weak-coupling regime, that is, when the inve…
View article: Strong renewal theorems and local large deviations for multivariate random walks and renewals
Strong renewal theorems and local large deviations for multivariate random walks and renewals Open
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 $\\boldsymbol{\\alpha } =(\\alpha _1,\\ldots ,\\alpha _d) \\in (0,2]^d$: in particular, we …