On the support of the simple branching random walk

Connectivity of the support of the simple branching random walk is established in certain asymmetric cases, extending a previous result of Grill.     

Occupation measure for random walk on the circle

We give bounds on the probability of deviation of the occupation measure of an interval on the circle for random walk.     

The ratio set of the harmonic measure of a random walk on a hyperbolic group

We consider the harmonic measure on the Gromov boundary of a non-amenable hyperbolic group defined by a finite range random walk on the group, and study the corresponding orbit equivalence relation on the boundary. It is known to be always amenable and of type III. We determine its ratio set by showing that it is generated by certain values of the Martin kernel. In particular, we show that the equivalence relation is never of type III₀.     

On the range of random walk

Let {S _n , n=0, 1, 2, …} be a random walk (S _n being thenth partial sum of a sequence of independent, identically distributed, random variables) with values inE _d , thed-dimensional integer lattice. Letf _n =Prob {S ₁ ≠ 0, …,S _n −1 ≠ 0,S _n =0 |S ₀=0}. The random walk is said to be transient if and strongly transient if . LetR _n =cardinality of the set {S ₀,S ₁, …,S _n }. It is shown that for a strongly transient random walk with p<1, the distribution of [R _n −np]/σ √n converges to the normal distribution with mean 0 and variance 1 asn tends to infinity, where σ is an appropriate positive constant. The other main result concerns the “capacity” of {S ₀, …,S _n }. For a finite setA inE _d , let C(A=Σ _x∈A ) Prob {S _n ∉A, n≧1 |S ₀=x} be the capacity ofA. A strong law forC{S ₀, …,S _n } is proved for a transient random walk, and some related questions are also considered. This research was partially supported by the National Science Foundation.     

Matrix random products with singular harmonic measure

Any Zariski dense countable subgroup of SL(d, mathbb R){SL(d, mathbb {R})} is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in mathbb R^d{mathbb {R}^d} in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and arbitrarily long Lyapunov vector.     

On the speed of a cookie random walk

We consider the model of the one-dimensional cookie random walk when the initial cookie distribution is spatially uniform and the number of cookies per site is finite. We give a criterion to decide whether the limiting speed of the walk is non-zero. In particular, we show that a positive speed may be obtained for just three cookies per site. We also prove a result on the continuity of the speed with respect to the initial cookie distribution.      

On a semi-Markov generalization of the random walk

The semi-Markov process studied here is a generalized random walk on the non-negative integers with zero as a reflecting barrier, in which the time interval between two consecutive jumps is given an arbitrary distribution H(t). Our process is identical with the Markov chain studied by Miller [6] in the special case when H(t)=U₁(t), the Heaviside function with unit jump at t=1. By means of a Spitzer-Baxter type identity, we establish criteria for transience, positive and null recurrence, as well as conditions for exponential ergodicity. The results obtained here generalize those of [6] and some classical results in random walk theory [10].     

On the trajectory of an individual chosen according to supercritical Gibbs measure in the branching random walk

Consider a branching random walk on the real line. Madaule (2016) showed the renormalized trajectory of an individual selected according to the critical Gibbs measure converges in law to a Brownian meander. Besides, Chen (2015) proved that the renormalized trajectory leading to the leftmost individual at time $n$ converges in law to a standard Brownian excursion. In this article, we prove that the renormalized trajectory of an individual selected according to a supercritical Gibbs measure also converges in law toward the Brownian excursion. Moreover, refinements of this results enables to express the probability for the trajectories of two individuals selected according to the Gibbs measure to have split before time $t$, partially answering a question of Derrida and Spohn (1988).     

On convergent probability of a random walk

This note introduces an interesting random walk on a straight path with cards of random numbers. The method of recurrent relations is used to obtain the convergent probability of the random walk with different initial positions.     

On the centre of mass of a random walk

For a random walk $S_n$ on ${\mathbb{R}}^d$ we study the asymptotic behaviour of the associated centre of mass process $G_n=n^{?1}{\sum }_{i=1}^n{S}_i$. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, $G_n$ is recurrent if $d=1$ and transient if $d\ge 2$. In the transient case we show that $G_n$ has a diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which $G_n$ is transient in $d=1$.     

On not storing the path of a random walk

We describe a novel form of Monte Carlo method with which to study self-avoiding random walks; we do not (in any sense) store the path of the walk being considered. As we show, the problem is related to that of devising a random-number generator which can produce itsnth number on request, without running through its sequence up to this point.     

Asymptotic properties of the occupation measure in a multidimensional skip-free Markov-modulated random walk

Queueing Systems - We consider a discrete-time d-dimensional process $$\{{\varvec{X}}_n\}=\{(X_{1,n},X_{2,n},\ldots ,X_{d,n})\}$$ on $${\mathbb {Z}}^d$$ with a background process $$\{J_n\}$$ on a...     

On the scaling limit of loop-erased random walk excursion

We use the known convergence of loop-erased random walk to radial SLE(2) to give a new proof that the scaling limit of loop-erased random walk excursion in the upper half-plane is chordal SLE(2). Our proof relies on a version of Wilson’s algorithm for weighted graphs which is used together with a Beurling-type estimate for random walk excursion. We also establish and use the convergence of the radial SLE path to the chordal SLE path as the bulk point tends to a boundary point. In the final section we sketch how to extend our results to more general simply connected domains.