Random walks in random environments on metric groups

Random walks in random environments on countable metric groups with bounded jumps of the walking particle are considered. The transition probabilities of such a random walk from a pointx εG (whereG is the group in question) are described by a vectorp(x) ε ℝ^|W| (whereW ⊏G is fixed and |W|<∞). The set {p(x),x εG} is assumed to consist of independent identically distributed random vectors. A sufficient condition for this random walk to be transient is found. As an example, the groups ℤ ^d , free groups, and the free product of finitely many cyclic groups of second order are considered. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 129–135, January, 2000.     

RENEWAL THEOREM FOR （L, 1）-RANDOM WALK IN RANDOM ENVIRONMENT

We consider a random walk on Z in random environment with possible jumps {-L,…, -1, 1}, in the case that the environment {ωi ： i ∈ Z} are i.i.d.. We establish the renewal theorem for the Markov chain of ＂the environment viewed from the particle＂ in both annealed probability and quenched probability, which generalize partially the results of Kesten （1977） and Lalley （1986） for the nearest random walk in random environment on Z, respectively. Our method is based on （L, 1）-RWRE formulated in Hong and Wang the intrinsic branching structure within the （2013）.     

Asymptotic behavior for random walk in random environment with holding times

In this article, we mainly discuss the asymptotic behavior for multi-dimensional continuous-time random walk in random environment with holding times. By constructing a renewal structure and using the point “environment viewed from the particle”, under General Kalikow's Condition, we show the law of large numbers (LLN) and central limit theorem (CLT) for the escape speed of random walk.     

Existence of a double random integral with respect to stable measures

In this paper, the authors study a double random integral of the form ∫₀¹∫₀¹f(s,t) M(ds) M(dt), where M(0,t) is a stable process with independent increments. Basically, the Wiener approach is used, and the existence of the above integral is established for a wide class of functions f.     

Component structure of the vacant set induced by a random walk on a random graph

We consider random walks on several classes of graphs and explore the likely structure of the vacant set, i.e. the set of unvisited vertices. Let Γ(t) be the subgraph induced by the vacant set of the walk at step t. We show that for random graphs G_n,p (above the connectivity threshold) and for random regular graphs G_r,r ≥ 3, the graph Γ(t) undergoes a phase transition in the sense of the well‐known ErdJW‐RSAT1100590x.png ‐Renyi phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique giant component, plus components of size O(log n), and for t ≥ (1 + ε)t^* all components are of size O(log n). For G_n,p and G_r we give the value of t^*, and the size of Γ(t). For G_r, we also give the degree sequence of Γ(t), the size of the giant component (if any) of Γ(t) and the number of tree components of Γ(t) of a given size k = O(log n). We also show that for random digraphs D_n,p above the strong connectivity threshold, there is a similar directed phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique strongly connected giant component, plus strongly connected components of size O(log n), and for t ≥ (1 + ε)t^* all strongly connected components are of size O(log n). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Level crossing probabilities II: Polygonal recurrence of multidimensional random walks

In part I we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is O(n−1/2). In higher dimensions we call a random walk ‘polygonally recurrent’ if there is a bounded set, hit by infinitely many of the straight lines between two consecutive sites a.s. The above estimate implies that three-dimensional random walks with independent components are polygonally transient. Similarly a directionally reinforced random walk on Z³ in the sense of Mauldin, Monticino and von Weizsäcker [R.D. Mauldin, M. Monticino, H. von Weizsäcker, Directionally reinforced random walks, Adv. Math. 117 (1996) 239-252] is transient. On the other hand, we construct an example of a transient but polygonally recurrent random walk with independent components on Z².     

Catalytic branching random walks in ℤ d with branching at the origin

A time-continuous branching random walk on the lattice ? ^d , d ≥ 1, is considered when the particles may produce offspring at the origin only. We assume that the underlying Markov random walk is homogeneous and symmetric, the process is initiated at moment t = 0 by a single particle located at the origin, and the average number of offspring produced at the origin is such that the corresponding branching random walk is critical. The asymptotic behavior of the survival probability of such a process at moment t → ∞ and the presence of at least one particle at the origin is studied. In addition, we obtain the asymptotic expansions for the expectation of the number of particles at the origin and prove Yaglom-type conditional limit theorems for the number of particles located at the origin and beyond at moment t.     

Multivariate operator-self-similar random fields

Multivariate random fields whose distributions are invariant under operator-scalings in both the time domain and the state space are studied. Such random fields are called operator-self-similar random fields and their scaling operators are characterized. Two classes of operator-self-similar stable random fields X={X(t),t∈R^d} with values in R^m are constructed by utilizing homogeneous functions and stochastic integral representations.     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.     

A variational approach to dislocation problems for periodic Schrödinger operators

As a simple model for lattice defects like grain boundaries in solid state physics we consider potentials which are obtained from a periodic potential V=V(x,y) on R² with period lattice Z² by setting Wt(x,y)=V(x+t,y) for x<0 and Wt(x,y)=V(x,y) for x?0, for t∈[0,1]. For Lipschitz-continuous V it is shown that the Schrödinger operators Ht=−Δ+Wt have spectrum (surface states) in the spectral gaps of H₀, for suitable t∈(0,1). We also discuss the density of these surface states as compared to the density of the bulk. Our approach is variational and it is first applied to the well-known dislocation problem (Korotyaev (2000, 2005) [15] and [16]) on the real line. We then proceed to the dislocation problem for an infinite strip and for the plane. In Appendix A, we discuss regularity properties of the eigenvalue branches in the one-dimensional dislocation problem for suitable classes of potentials.     

ASYMPTOTIC BEHAVIOR FOR RANDOM WALK IN RANDOM ENVIRONMENT WITH HOLDING TIMES

In this article, we mainly discuss the asymptotic behavior for multi-dimensional continuous-time random walk in random environment with holding times. By constructing a renewal structure and using the point ＂environment viewed from the particle＂, under General Kalikow＇s Condition, we show the law of large numbers （LLN） and central limit theorem （CLT） for the escape speed of random walk.     

Transient random walks in random environment on a Galton–Watson tree

We consider a transient random walk (X _n ) in random environment on a Galton–Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be positive. As a consequence, situations with zero speed are revealed to occur. In such cases, we prove that X _n is of order of magnitude n ^Λ, with ${\Lambda \in (0,1)}$ . We also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed, which improves a recent result of Collevecchio (Probab Theory Related 136(1):81–101, 2006).     

Hitting times with taboo for a random walk

For a symmetric homogeneous and irreducible random walk on the d-dimensional integer lattice, which have a finite variance of jumps, we study passage times (taking values in [0,??]) determined by a starting point x, a hitting state y, and a taboo state z. We find the probability that these passage times are finite, and study the distribution tail. In particular, it turns out that, for the above-mentioned random walks on ? ^d except for a simple random walk on ?, the order of the distribution tail decrease is specified by dimension d only. In contrast, for a simple random walk on ?, the asymptotic properties of hitting times with taboo essentially depend on mutual location of the points x, y, and z. These problems originated in recent study of a branching random walk on ? ^d with a single source of branching.     

Asymptotic behavior of a stochastic growth process associated with a system of interacting branching random walksComportement asymptotique d'un processus stochastique de croissance associé à un système de marches aléatoires en interaction

We study a continuous time growth process on $\text{Z}{}^{\mathrm{d}}$ (d?1) associated to the following interacting particle system: initially there is only one simple symmetric continuous time random walk of total jump rate one located at the origin; then, whenever a random walk visits a site still unvisited by any other random walk, it creates a new independent random walk starting from that site. Let us call P_d the law of such a process and S⁰_d(t) the set of sites, visited by all walks by time t. We prove that there exists a bounded, non-empty, convex set $\text{C}{}_{\mathrm{d}}\text{?}\text{R}{}^{\mathrm{d}}$, such that for every ε>0, P_d-a.s. eventually in t, the set S_d⁰(t) is within an ε neighborhood of the set [C_dt], where for $\text{A?}\text{R}{}^{\mathrm{d}}$ we define $\text{[A]:=A∩}\text{Z}{}^{\mathrm{d}}$. Moreover, for d large enough, the set C_d is not a ball under the Euclidean norm. We also show that the empirical density of particles within S_d⁰(t) converges weakly to a product Poisson measure of parameter one. To cite this article: A.F. Ram??rez, V. Sidoravicius, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 821–826.     

Limiting behavior of a diffusion in an asymptotically stable environment

Let V be a two sided random walk and let X denote a real valued diffusion process with generator . This process is the continuous equivalent of the one-dimensional random walk in random environment with potential V. Hu and Shi (1997) described the Lévy classes of X in the case where V behaves approximately like a Brownian motion. In this paper, based on some fine results on the fluctuations of random walks and stable processes, we obtain an accurate image of the almost sure limiting behavior of X when V behaves asymptotically like a stable process. These results also apply for the corresponding random walk in random environment.     

An L0（F,R）-valued Function＇s Intermediate Value Theorem and Its Applications to Random Uniform Convexity

Let (Ω , F , P ) be a probability space and L0 ( F, R ) the algebra of equivalence classes of real- valued random variables on (Ω , F , P ). When L0 ( F, R ) is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0 ( F, R ) to L0 ( F, R ). As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module ( S,|| · ||) is random uniformly convex iff Lp ( S ) is uniformly convex for each fixed positive number p such that 1 p + ∞ .     

On differential inclusions with prescribed attainable sets

In this article, the inverse problem of the differential inclusion theory is studied. For a given ε>0 and a continuous set valued map t→W(t), t∈[t₀,θ], where W(t)⊂Rⁿ is compact and convex for every t∈[t₀,θ], it is required to define differential inclusion so that the Hausdorff distance between the attainable set of the differential inclusion at the time moment t with initial set (t₀,W(t₀)) and W(t) would be less than ε for every t∈[t₀,θ].     

New Partial Difference Sets in Ztp2 and a Related Problem about Galois Rings

We generalize a construction of partial difference sets (PDS) by Chen, Ray-Chaudhuri, and Xiang through a study of the Teichmüller sets of the Galois rings. Let R=GR(p², t) be the Galois ring of characteristic p² and rank t with Teichmüller set T and let π:R→R/pR be the natural homomorphism. We give a construction of PDS in R with the parameters ν=p^2t, k=r(p^t−1), λ=p^t+r²−3r, μ=r²−r, where r=lp^{t−s(p, t)}, 1≤l≤p^{s(p, t)}, and s(p, t) is the largest dimension of a GF(p)-subspace W⊂R/pR such that π⁻¹(W)∩T generates a subgroup of R of rank <t. We prove that s(p, t) is the largest dimension of a GF(p)-subspace W of GF(p^t) such that dim W^p<t, where W^p is the GF(p)-space generated by {∏^p_i=1w_i∣w_i∈W, 1≤i≤p}. We determine the values of s(p, t) completely and solve a general problem about dim_EW^r for an E-vector space W in a finite extension of a finite field E. The PDS constructed here contain the family constructed by Chen, Ray-Chaudhuri, and Xiang and have a wider range of parameters.     

Loop-Erased Random Walk on Finite Graphs and the Rayleigh Process

Let (G _n ) _n=1^∞ be a sequence of finite graphs, and let Y _t be the length of a loop-erased random walk on G _n after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which G _n is the d-dimensional torus of size-length n for d≥4, the process (Y _t ) _t=0^∞, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. Our proof relies heavily on ideas of Peres and Revelle, who used loop-erased random walks to show that the uniform spanning tree on large finite graphs converges to the Brownian continuum random tree of Aldous. Supported in part by NSF Grant DMS-0504882.     

Random walk in mixed random environment without uniform ellipticity

We study a random walk in random environment on ?₊. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i) points endowed with probabilities drawn from a symmetric distribution with heavy tails at 0 and 1, and (ii) “fast points” with a fixed systematic drift. Without these fast points, the model is related to the diffusion in heavy-tailed (“stable”) random potential studied by Schumacher and Singh; the fast points perturb that model. The two components compete to determine the behaviour of the random walk; we identify phase transitions in terms of the model parameters. We give conditions for recurrence and transience and prove almost sure bounds for the trajectories of the walk.