arXiv subjects

Rongfeng Sun

Publications and source records attributed to Rongfeng Sun.

50 records · Page 3Linked to original sources

arXiv2013

Symmetric Rearrangements Around Infinity with Applications to Levy Processes

We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger (1976) and can be interpreted as involving symmetric rearrangements of domains around infinity. As applications, we prove two comparison results for general Levy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Levy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Levy motions are symmetrically rearranged. This essentially generalizes an isoperimetric inequality of Peres and Sousi (2011) for the Wiener sausage. In the second application, we show that the q-capacity of a Borel measurable set for a Levy process can only decrease if the set and the Levy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe (1983) for symmetric Levy processes.

math.PR↗

arXiv2012

A Monotonicity Result for the Range of a Perturbed Random Walk

We consider a discrete time simple symmetric random walk on Z^d, d>=1, where the path of the walk is perturbed by inserting deterministic jumps. We show that for any time n and any deterministic jumps that we insert, the expected number of sites visited by the perturbed random walk up to time n is always larger than or equal to that for the unperturbed walk. This intriguing problem arises from the study of a particle among a Poisson system of moving traps with sub-diffusive trap motion. In particular, our result implies a variant of the Pascal principle, which asserts that among all deterministic trajectories the particle can follow, the constant trajectory maximizes the particle's survival probability up to any time t>0.

math.PR↗

arXiv2012

Survival Probability of a Random Walk Among a Poisson System of Moving Traps

We review some old and prove some new results on the survival probability of a random walk among a Poisson system of moving traps on Z^d, which can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential. We show that the annealed survival probability decays asymptotically as e^{-λ_1\sqrt{t}} for d=1, as e^{-λ_2 t/\log t} for d=2, and as e^{-λ_d t} for d>= 3, where λ_1 and λ_2 can be identified explicitly. In addition, we show that the quenched survival probability decays asymptotically as e^{-\tilde λ_d t}, with \tilde λ_d>0 for all d>= 1. A key ingredient in bounding the annealed survival probability is what is known in the physics literature as the Pascal principle, which asserts that the annealed survival probability is maximized if the random walk stays at a fixed position. A corollary of independent interest is that the expected cardinality of the range of a continuous time symmetric random walk increases under perturbation by a deterministic path.

math.PR↗

arXiv2012

A monotonicity property for random walk in a partially random environment

We prove a law of large numbers for random walks in certain kinds of i.i.d. random environments in Z^d that is an extension of a result of Bolthausen, Sznitman and Zeitouni (2003). We use this result, along with the lace expansion for self-interacting random walks, to prove a monotonicity result for the first coordinate of the speed of the random walk under some strong assumptions on the distribution of the environment.

math.PR↗

arXiv2011

One-dimensional Voter Model Interface Revisited

We consider the voter model on Z, starting with all 1's to the left of the origin and all 0's to the right of the origin. It is known that if the associated random walk kernel p has zero mean and a finite r-th moment for any r>3, then the evolution of the boundaries of the interface region between 1's and 0's converge in distribution to a standard Brownian motion (B_t)_{t>0} under diffusive scaling of space and time. This convergence fails when p has an infinite r-th moment for any r<3, due to the loss of tightness caused by a few isolated 1's appearing deep within the regions of all 0's (and vice versa) at exceptional times. In this note, we show that as long as p has a finite second moment, the measure-valued process induced by the rescaled voter model configuration is tight, and converges weakly to the measure-valued process 1_{x 0.

math.PR↗

arXiv2011

Brownian Web and Oriented Percolation: Density Bounds

In a recent work, we proved that under diffusive scaling, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice Z^2 converges in distribution to the Brownian web. In that proof, the FKG inequality played an important role in establishing a density bound, which is a part of the convergence criterion for the Brownian web formulated by Fontes et al (2004). In this note, we illustrate how an alternative convergence criterion formulated by Newman et al (2005) can be verified in this case, which involves a dual density bound that can be established without using the FKG inequality. This alternative approach is in some sense more robust. We will also show that the spatial density of the collection of rightmost infinite open paths starting at time 0 decays asymptotically in time as c/\sqrt{t} for some c>0.

math.PR↗

arXiv2010

Disorder relevance for the random walk pinning model in dimension 3

We study the continuous time version of the random walk pinning model, where conditioned on a continuous time random walk Y on Z^d with jump rate ρ>0, which plays the role of disorder, the law up to time t of a second independent random walk X with jump rate 1 is Gibbs transformed with weight e^{βL_t(X,Y)}, where L_t(X,Y) is the collision local time between X and Y up to time t. As the inverse temperature βvaries, the model undergoes a localization-delocalization transition at some critical β_c>=0. A natural question is whether or not there is disorder relevance, namely whether or not β_c differs from the critical point β_c^{ann} for the annealed model. In Birkner and Sun [BS09], it was shown that there is disorder irrelevance in dimensions d=1 and 2, and disorder relevance in d>=4. For d>=5, disorder relevance was first proved by Birkner, Greven and den Hollander [BGdH08]. In this paper, we prove that if X and Y have the same jump probability kernel, which is irreducible and symmetric with finite second moments, then there is also disorder relevance in the critical dimension d=3, and β_c-β^{ann}_c is at least of the order e^{-C(ζ)ρ^{-ζ}}, C(ζ)>0, for any ζ>2. Our proof employs coarse graining and fractional moment techniques, which have recently been applied by Lacoin [L09] to the directed polymer model in random environment, and by Giacomin, Lacoin and Toninelli [GLT09] to establish disorder relevance for the random pinning model in the critical dimension. Along the way, we also prove a continuous time version of Doney's local limit theorem [D97] for renewal processes with infinite mean.

math.PR↗

arXiv2009

Annealed vs Quenched Critical Points for a Random Walk Pinning Model

We study a random walk pinning model, where conditioned on a simple random walk Y on Z^d acting as a random medium, the path measure of a second independent simple random walk X up to time t is Gibbs transformed with Hamiltonian -L_t(X,Y), where L_t(X,Y) is the collision local time between X and Y up to time t. This model arises naturally in various contexts, including the study of the parabolic Anderson model with moving catalysts, the parabolic Anderson model with Brownian noise, and the directed polymer model. It falls in the same framework as the pinning and copolymer models, and exhibits a localization-delocalization transition as the inverse temperature βvaries. We show that in dimensions d=1,2, the annealed and quenched critical values of βare both 0, while in dimensions d\geq 4, the quenched critical value of βis strictly larger than the annealed critical value (which is positive). This implies the existence of certain intermediate regimes for the parabolic Anderson model with Brownian noise and the directed polymer model. For d\geq 5, the same result has recently been established by Birkner, Greven and den Hollander via a quenched large deviation principle. Our proof is based on a fractional moment method used recently by Derrida, Giacomin, Lacoin and Toninelli to establish the non-coincidence of annealed and quenched critical points for the pinning model in the disorder-relevant regime. The critical case d=3 remains open.

math.PR↗

arXiv2008

Special points of the Brownian net

The Brownian net, which has recently been introduced by Sun and Swart [SS08], and independently by Newman, Ravishankar and Schertzer [NRS08], generalizes the Brownian web by allowing branching. In this paper, we study the structure of the Brownian net in more detail. In particular, we give an almost sure classification of each point in $R^2$ according to the configuration of the Brownian net paths entering and leaving the point. Along the way, we establish various other structural properties of the Brownian net.

math.PR↗

arXiv2008

The Brownian net

The (standard) Brownian web is a collection of coalescing one- dimensional Brownian motions, starting from each point in space and time. It arises as the diffusive scaling limit of a collection of coalescing random walks. We show that it is possible to obtain a nontrivial limiting object if the random walks in addition branch with a small probability. We call the limiting object the Brownian net, and study some of its elementary properties.

math.PR↗

arXiv2008

A quenched limit theorem for the local time of random walks on \Z^2

Let $X$ and $Y$ be two independent random walks on $\Z^2$ with zero mean and finite variances, and let $L_t(X,Y)$ be the local time of $X-Y$ at the origin at time $t$. We show that almost surely with respect to $Y$, $L_t(X,Y)/\log t$ conditioned on $Y$ converges in distribution to an exponential random variable with the same mean as the distributional limit of $L_t(X,Y)/\log t$ without conditioning. This question arises naturally from the study of the parabolic Anderson model with a single moving catalyst, which is closely related to a pinning model.

math.PR↗

arXiv2007

The renormalization transformation for two-type branching models

This paper studies countable systems of linearly and hierarchically interacting diffusions taking values in the positive quadrant. These systems arise in population dynamics for two types of individuals migrating between and interacting within colonies. Their large-scale space-time behavior can be studied by means of a renormalization program. This program, which has been carried out successfully in a number of other cases (mostly one-dimensional), is based on the construction and the analysis of a nonlinear renormalization transformation, acting on the diffusion function for the components of the system and connecting the evolution of successive block averages on successive time scales. We identify a general class of diffusion functions on the positive quadrant for which this renormalization transformation is well-defined and, subject to a conjecture on its boundary behavior, can be iterated. Within certain subclasses, we identify the fixed points for the transformation and investigate their domains of attraction. These domains of attraction constitute the universality classes of the system under space-time scaling.

math.PR↗

arXiv2005

Convergence of Coalescing Nonsimple Random Walks to the Brownian Web

The Brownian Web (BW) is a family of coalescing Brownian motions starting from every point in space and time $\R\times\R$. It was first introduced by Arratia, and later analyzed in detail by Tóth and Werner. More recently, Fontes, Isopi, Newman and Ravishankar gave a characterization of the BW, and general convergence criteria allowing either crossing or noncrossing paths, which they verified for coalescing simple random walks. Later Ferrari, Fontes, and Wu verified these criteria for a two dimensional Poisson Tree. In both cases, the paths are noncrossing. In this thesis, we formulate new convergence criteria for crossing paths, and verify them for non-simple coalescing random walks (both discrete and continuous time) satisfying a finite fifth moment condition. This is the first time convergence to the BW has been proved for models with crossing paths. Several corollaries are presented, including an analysis of the scaling limit of voter model interfaces that extends a result of Cox and Durrett.

math.PR↗

arXiv2000

Cluster Monte Carlo study of multi-component fluids of the Stillinger-Helfand and Widom-Rowlinson type

Phase transitions of fluid mixtures of the type introduced by Stillinger and Helfand are studied using a continuum version of the invaded cluster algorithm. Particles of the same species do not interact, but particles of different types interact with each other via a repulsive potential. Examples of interactions include the Gaussian molecule potential and a repulsive step potential. Accurate values of the critical density, fugacity and magnetic exponent are found in two and three dimensions for the two-species model. The effect of varying the number of species and of introducing quenched impurities is also investigated. In all the cases studied, mixtures of $q$-species are found to have properties similar to $q$-state Potts models.

cond-mat.stat-mech↗