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Yueyun Hu

Publications and source records attributed to Yueyun Hu.

51 records · Page 3Linked to original sources
arXiv2011

Almost sure convergence for stochastically biased random walks on trees

We are interested in the biased random walk on a supercritical Galton--Watson tree in the sense of Lyons, Pemantle and Peres, and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system's non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)^3, converges almost surely to a known positive constant.

math.PR↗
arXiv2010

Charged polymers in the attractive regime: a first order transition from Brownian scaling to four points localization

We study a quenched charged-polymer model, introduced by Garel and Orland in 1988, that reproduces the folding/unfolding transition of biopolymers. We prove that, below the critical inverse temperature, the polymer is delocalized in the sense that: (1) The rescaled trajectory of the polymer converges to the Brownian path; and (2) The partition function remains bounded. At the critical inverse temperature, we show that the maximum time spent at points jumps discontinuously from 0 to a positive fraction of the number of monomers, in the limit as the number of monomers tends to infinity. Finally, when the critical inverse temperature is large, we prove that the polymer collapses in the sense that a large fraction of its monomers live on four adjacent positions, and its diameter grows only logarithmically with the number of the monomers. Our methods also provide some insight into the annealed phase transition and at the transition due to a pulling force; both phase transitions are shown to be discontinuous.

math.PR↗
arXiv2010

Branching random walk in Z^4 with branching at the origin only

For the critical branching random walk in $\mathbb{Z}^{4}$ with branching at the origin only we find the asymptotic behavior of the probability of the event that there are particles at the origin at moment $t\rightarrow \infty $ and prove a Yaglom type conditional limit theorem for the number of individuals at the origin given that there are particles at the origin.

math.PR↗
arXiv2010

Asymptotics for the survival probability in a killed branching random walk

Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope $γ-ε$, where $γ$ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when $ε\to 0$, the probability in question decays like $\exp\{- {β+ o(1)\over ε^{1/2}}\}$, where $β$ is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli$(p)$ random variables (with $0<p<{1\over 2}$) assigned on a rooted binary tree, this answers an open question of Robin Pemantle.

math.PR↗
arXiv2009

Strong approximations in a charged-polymer model

We study the large-time behavior of the charged-polymer Hamiltonian $H_n$ of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths, and Higgs [Gaussian case], using strong approximations to Brownian motion. Our results imply, among other things, that in one dimension the process $\{H_{[nt]}\}_{0\le t\le 1}$ behaves like a Brownian motion, time-changed by the intersection local-time process of an independent Brownian motion. Chung-type LILs are also discussed.

math.PR↗
arXiv2009

Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees

We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou [9]. Our method applies furthermore to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [17]. Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

math.PR↗
arXiv2007

Localization for Branching Random Walks in Random Environment

We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If $d \ge 3$ and the environment is "not too random", then, the total population grows as fast as its expectation with strictly positive probability. If,on the other hand, $d \le 2$, or the environment is ``random enough", then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of "replica overlap". We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.

math.PR↗
arXiv2006

Slow movement of random walk in random environment on a regular tree

We consider a recurrent random walk in random environment on a regular tree. Under suitable general assumptions upon the distribution of the environment, we show that the walk exhibits an unusual slow movement: the order of magnitude of the walk in the first $n$ steps is $(\log n)^3$.

math.PR↗
arXiv2006

A subdiffusive behaviour of recurrent random walk in random environment on a regular tree

We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle [11] give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk $(X\_n)$ in random environment on a regular tree, which is closely related to Mandelbrot [13]'s multiplicative cascade. We prove, under some general assumptions upon the distribution of the environment, the existence of a new exponent $ν\in (0, {1\over 2}]$ such that $\max\_{0\le i \le n} |X\_i|$ behaves asymptotically like $n^ν$. The value of $ν$ is explicitly formulated in terms of the distribution of the environment.

math.PR↗
arXiv2005

Moderate deviations for diffusions with Brownian potentials

We present precise moderate deviation probabilities, in both quenched and annealed settings, for a recurrent diffusion process with a Brownian potential. Our method relies on fine tools in stochastic calculus, including Kotani's lemma and Lamperti's representation for exponential functionals. In particular, our result for quenched moderate deviations is in agreement with a recent theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571-609] who studied the corresponding problem for Sinai's random walk in random environment.

math.PR↗
arXiv2005

On the increments of the principal value of Brownian local time

Let $W$ be a one-dimensional Brownian motion starting from 0. Define $Y(t)= \int_0^t{\d s \over W(s)} := \lim_{ε\to0} \int_0^t 1_{(|W(s)|> ε)} {\d s \over W(s)} $ as Cauchy's principal value related to local time. We prove limsup and liminf results for the increments of $Y$.

math.PR↗
arXiv2004

Strong approximations of three-dimensional Wiener sausages

In this paper we prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall's estimates between the Wiener sausage and the Brownian intersection local times.

math.PR↗