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Yueyun Hu

Publications and source records attributed to Yueyun Hu.

At least 37 records · Page 2Linked to original sources
arXiv2020

The Derrida--Retaux conjecture on recursive models

We are interested in the nearly supercritical regime in a family of max-type recursive models studied by Collet, Eckman, Glaser and Martin and by Derrida and Retaux, and prove that under a suitable integrability assumption on the initial distribution, the free energy vanishes at the transition with an essential singularity with exponent $\tfrac12$. This gives a weaker answer to a conjecture of Derrida and Retaux. Other behaviours are obtained when the integrability condition is not satisfied.

math.PR↗
arXiv2019

Large deviations for level sets of branching Brownian motion and Gaussian free fields

We study deviation probabilities for the number of high positioned particles in branching Brownian motion, and confirm a conjecture of Derrida and Shi (2016). We also solve the corresponding problem for the two-dimensional discrete Gaussian free field. Our method relies on an elementary inequality for inhomogeneous Galton-Watson processes.

math.PR↗
arXiv2019

A max-type recursive model: some properties and open questions

We consider a simple max-type recursive model which was introduced in the study of depinning transition in presence of strong disorder, by Derrida and Retaux. Our interest is focused on the critical regime, for which we study the extinction probability, the first moment and the moment generating function. Several stronger assertions are stated as conjectures.

math.PR↗
arXiv2019

An exactly solvable continuous-time Derrida--Retaux model

To study the depinning transition in the limit of strong disorder, Derrida and Retaux (2014) introduced a discrete-time max-type recursive model. It is believed that for a large class of recursive models, including Derrida and Retaux' model, there is a highly non-trivial phase transition. In this article, we present a continuous-time version of Derrida and Retaux model, built on a Yule tree, which yields an exactly solvable model belonging to this universality class. The integrability of this model allows us to study in details the phase transition near criticality and can be used to confirm the infinite order phase transition predicted by physicists. We also study the scaling limit of this model at criticality, which we believe to be universal.

math.PR↗
arXiv2019

Local times of subdiffusive biased walks on trees

Consider a class of null-recurrent randomly biased walks on a super-critical Gaton-Watson tree. We obtain the rates of convergence of the local times and the quenched local probability for the biased walk in the sub-diffusive case. These results are a consequence of a sharp estimate on the return time, whose analysis is driven by a family of concave recursive equations on trees.

math.PR↗
arXiv2018

Resistance growth of branching random networks

Consider a rooted infinite Galton-Watson tree with mean offspring number $m>1$, and a collection of i.i.d. positive random variables $ξ_e$ indexed by all the edges in the tree. We assign the resistance $m^d ξ_e$ to each edge $e$ at distance $d$ from the root. In this random electric network, we study the asymptotic behavior of the effective resistance and conductance between the root and the vertices at depth $n$. Our results generalize an existing work of Addario-Berry, Broutin and Lugosi on the binary tree to random branching networks.

math.PR↗
arXiv2018

The free energy in the Derrida--Retaux recursive model

We are interested in a simple max-type recursive model studied by Derrida and Retaux (2014) in the context of a physics problem, and find a wide range for the exponent in the free energy in the nearly supercritical regime.

math.PR↗
arXiv2017

How big is the minimum of a branching random walk?

Let $M_n$ be the minimal position at generation $n$, of a real-valued branching random walk in the boundary case. As $n \to \infty$, $M_n- {3 \over 2} \log n$ is tight (see [1][9][2]). We establish here a law of iterated logarithm for the upper limits of $M_n$: upon the system's non-extinction, $ \limsup\_{n\to \infty} {1\over \log \log \log n} ( M_n - {3\over2} \log n) = 1$ almost surely. We also study the problem of moderate deviations of $M_n$: $p(M_n- {3 \over 2} \log n > λ)$ for $λ\to \infty$ and $λ=o(\log n)$. This problem is closely related to the small deviations of a class of Mandelbrot's cascades.

math.PR↗
arXiv2016

Favorite sites of randomly biased walks on a supercritical Galton-Watson tree

Erdős and Révész initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Gaton-Watson tree. We prove that there is some parameter $κ\in (1, \infty]$ such that the set of the favorite sites of the biased walk is almost surely bounded in the case $κ\in (2, \infty]$, tight in the case $κ=2$, and oscillates between a neighborhood of the root and the boundary of the range in the case $κ\in (1, 2)$. Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case $κ\in (2, \infty]$. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton-Watson trees.

math.PR↗
arXiv2016

The maximal drawdown of the Brownian meander

Motivated by evaluating the limiting distribution of randomly biased random walks on trees, we compute the exact value of a negative moment of the maximal drawdown of the standard Brownian meander.

math.PR↗
arXiv2015

The slow regime of randomly biased walks on trees

We are interested in the randomly biased random walk on the supercritical Galton--Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_n)$ is null recurrent, making a maximal displacement of order of magnitude $(\log n)^3$ in the first $n$ steps. We study the localization problem of $X_n$ and prove that the quenched law of $X_n$ can be approximated by a certain invariant probability depending on $n$ and the random environment. As a consequence, we establish that upon the survival of the system, $\frac{|X_n|}{(\log n)^2}$ converges in law to some non-degenerate limit on $(0, \infty)$ whose law is explicitly computed.

math.PR↗
arXiv2015

The most visited sites of biased random walks on trees

We consider the slow movement of randomly biased random walk $(X_n)$ on a supercritical Galton--Watson tree, and are interested in the sites on the tree that are most visited by the biased random walk. Our main result implies tightness of the distributions of the most visited sites under the annealed measure. This is in contrast with the one-dimensional case, and provides, to the best of our knowledge, the first non-trivial example of null recurrent random walk whose most visited sites are not transient, a question originally raised by Erdős and Révész [11] for simple symmetric random walk on the line.

math.PR↗
arXiv2014

The minimum of a branching random walk outside the boundary case

This paper is a complement to the studies on the minimum of a real-valued branching random walk. In the boundary case (Biggins, Kyprianou 2005), Aïdékon in a seminal paper (2013) obtained the convergence in law of the minimum after a suitable renormalization. We study here the situation when the log-generating function of the branching random walk explodes at some positive point and it cannot be reduced to the boundary case. In the associated thermodynamics framework this corresponds to a first order phase transition, while the boundary case corresponds to a second order phase transition.

math.PR↗
arXiv2013

The precise tail behavior of the total progeny of a killed branching random walk

Consider a branching random walk on the real line with a killing barrier at zero: starting from a nonnegative point, particles reproduce and move independently, but are killed when they touch the negative half-line. The population of the killed branching random walk dies out almost surely in both critical and subcritical cases, where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed, and by critical case, when this speed is zero. We investigate the total progeny of the killed branching random walk and give their precise tail distribution both in the critical and subcritical cases, which solves an open problem of Aldous [Power laws and killed branching random walks, http://www.stat.berkeley.edu/~aldous/Research/OP/brw.html].

math.PR↗
arXiv2012

The Spread of a Catalytic Branching Random Walk

We consider a random walk on $\Z$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_n$. Then we determine all possible limiting law for the sequence $M_n -αn$ where $α$ is a deterministic constant.

math.PR↗
arXiv2011

Einstein relation for biased random walk on Galton--Watson trees

We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton--Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrary deep traps.

math.PR↗