Quantum Bernoulli noises are annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relations (CAR) in equal-time. In this paper, we use quantum Bernoulli noises to introduce a model of open quantum walk on the $d$-dimensional integer lattice $\mathbb{Z}^d$ for a general positive integer $d\geq 2$, which we call the $d$-dimensional open QBN walk. We obtain a quantum channel representation of the $d$-dimensional open QBN walk, and find that it admits the ``separability-preserving'' property. We prove that, for a wide range of choices of its initial state, the $d$-dimensional open QBN walk has a limit probability distribution of $d$-dimensional Gauss type. Finally we unveil links between the $d$-dimensional open QBN walk and the unitary quantum walk recently introduced in [Ce Wang and Caishi Wang, Higher-dimensional quantum walk in terms of quantum Bernoulli noises, Entropy 2020, 22, 504].
We consider a random walk among i.i.d. obstacles on the one dimensional integer lattice under the condition that the walk starts from the origin and reaches a remote location y. The obstacles are represented by a killing potential, which takes value M>0 with probability p and value 0 with probability (1-p), 0<p<1, independently at each site of the lattice. We consider the walk under both quenched and annealed measures. It is known that under either measure the crossing time from 0 to y of such walk, tau(y), grows linearly in y. More precisely, the expectation of tau(y)/y converges to a limit a…
We study a large class of long-range random walks which take values on the vertices of an N dimensional hypercube. These processes are connected with multivariate Bernoulli autoregression.
We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone is $O(1/ε^2)$, where $ε$ is the step size. Our main results show that for a class of Bernoulli maps, when the random walk is alternated with the Bernoulli map $φ$ the mixing time becomes $O(|\ln ε|)$. We also study the \emph{dissipation time} of this process, and obtain $O(|\ln ε|)$ upper and lower bounds with explicit constants.
We study some properties of the local time of the asymmetric Bernoulli walk on the line. These properties are very similar to the corresponding ones of the simple symmetric random walks in higher ($d\geq3$) dimension, which we established in the recent years. The goal of this paper is to highlight these similarities.
In this paper, we present a quantum Bernoulli noises approach to quantum walks on hypercubes. We first obtain an alternative description of a general hypercube and then, based on the alternative description, we find that the operators $\partial_k^* + \partial_k$ behave actually as the shift operators, where $\partial_k$ and $\partial_k^*$ are the annihilation and creation operators acting on Bernoulli functionals, respectively. With the above operators as the shift operators on the position space, we introduce a discrete-time quantum walk model on a general hypercube and obtain an explicit for…
We consider the use of random walks as an approach to obtain connection coefficients for higher-order Bernoulli and Euler polynomials. In particular, we consider the cases of a $1$-dimensional linear reflected Brownian motion and of a $3$-dimensional Bessel process. Considering the successive hitting times of two, three, and four fixed levels by these random walks yields non-trivial identities that involve higher-order Bernoulli and Euler polynomials.
We consider the simple random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ ($d \geq 1$), traveling in potentials which are Bernoulli distributed. The so-called Lyapunov exponent describes the cost of traveling for the simple random walk in the potential, and it is known that the Lyapunov exponent is strictly monotone in the parameter of the Bernoulli distribution. Hence, the aim of this paper is to investigate the effect of the potential on the Lyapunov exponent more precisely, and we derive some Lipschitz-type estimates for the difference between the Lyapunov exponents.
In this paper, we present an abstract model of continuous-time quantum walk (CTQW) based on Bernoulli functionals and show that the model has perfect state transfer (PST), among others. Let $\mathfrak{h}$ be the space of square integrable complex-valued Bernoulli functionals, which is infinitely dimensional. First, we construct on a given subspace $\mathfrak{h}_L \subset \mathfrak{h}$ a self-adjoint operator $Δ_L$ via the canonical unitary involutions on $\mathfrak{h}$, and by analyzing its spectral structure we find out all its eigenvalues. Then, we introduce an abstract model of CTQW with $\…
Let $(S_k)_{k\ge 1}$ be the classical Bernoulli random walk on the integer line with jump parameters $p\in(0,1)$ and $q=1-p$. The probability distribution of the sojourn time of the walk in the set of non-negative integers up to a fixed time is well-known, but its expression is not simple. By modifying slightly this sojourn time--through a particular counting process of the zeros of the walk as done by Chung & Feller ["On fluctuations in coin-tossings", Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 605-608]-, simpler representations may be obtained for its probability distribution. In the aforementi…
We consider a discrete time biased random walk conditioned to avoid Bernoulli obstacles on ${\mathbb Z}^d$ ($d\geq 2$) up to time $N$. This model is known to undergo a phase transition: for a large bias, the walk is ballistic whereas for a small bias, it is sub-ballistic. We prove that in the sub-ballistic phase, the random walk is contained in a ball of radius $O(N^{1/(d+2)})$, which is the same scale as for the unbiased case. As an intermediate step, we also prove large deviation principles for the endpoint distribution for the unbiased random walk at scales between $N^{1/(d+2)}$ and $o(N^{d…
A Bernoulli random walk is a random trajectory starting from 0 and having i.i.d. increments, each of them being $+1$ or -1, equally likely. The other families cited in the title are Bernoulli random walks under various conditionings. A peak in a trajectory is a local maximum. In this paper, we condition the families of trajectories to have a given number of peaks. We show that, asymptotically, the main effect of setting the number of peaks is to change the order of magnitude of the trajectories. The counting process of the peaks, that encodes the repartition of the peaks in the trajectories, i…
We consider the $1$-dimensional reflected Brownian motion and $3$-dimensional Bessel process and the general models. By decomposing the hitting times of consecutive sites into loops, we obtain identities, called loop identities, for the generating functions of the hitting times. After proving this decomposition both combinatorially and inductively, we consider the case that sites are equally distributed. Then, from loop identities, we derive expressions of Bernoulli and Euler polynomials, in terms of Euler polynomials of higher-orders.
The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) and their adjoint operators satisfy a canonical anti-commutation relation (CAR) in equal-time. As a mathematical structure, Dirichlet forms play an important role in many fields in mathematical physics. In this paper, we apply the Bernoulli annihilators to constructing Dirichlet forms on Bernoulli functionals. Let $w$ be a nonnegative function on $\mathbb{N}$. By using the Bernoulli annihilators, we first define in a dense subspace of the $L^2$-space of Bernoulli functionals a positive, symmetric bilinear f…
We consider a discrete time simple symmetric random walk among Bernoulli obstacles on $\mathbb{Z}^d$, $d\geq 2$, where the walk is killed when it hits an obstacle. It is known that conditioned on survival up to time $N$, the random walk range is asymptotically contained in a ball of radius $\varrho_N=C N^{1/(d+2)}$ for any $d\geq 2$. For $d=2$, it is also known that the range asymptotically contains a ball of radius $(1-ε)\varrho_N$ for any $ε>0$, while the case $d\geq 3$ remains open. We complete the picture by showing that for any $d\geq 2$, the random walk range asymptotically contains a ba…
We show quenched large deviations for the simple random walk on a certain class of percolations with long-range correlations. This class contains the supercritical Bernoulli percolations, the model considered by Drewitz, R'ath and Sapozhnikov and the random-cluster model up to the slab critical point. Our result is an extension of Kubota's result for the supercritical Bernoulli percolations. We also state a shape theorem for the chemical distance, which is an extension of Garet and Marchand's result for the supercritical Bernoulli percolations.
Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$ and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. For $d \geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment such that the origin is contained in an infinite connected component free of obstacles. It has previously been shown that with high probability, the random walk conditioned on survival up to time $n$ will be localized in a ball of volume asymptotically $d\log_{1/p}n$. In this work, we prove that this ba…
We study an extension of the generalized excited random walk (GERW) on $\mathbb{Z}^d$ introduced in [Ann. Probab. 40 (5), 2012, [7]] by Menshikov, Popov, Ramírez and Vachkovskaia. Our extension consists in studying a version of the GERW where excitation depends on a random environment. Given $p \in (0,1]$ (a parameter of the model) whenever the process visits a site for the first time, with probability $p$ it gains a drift in a given direction (could be any direction of the unit sphere). Otherwise, with probability $1-p$, it behaves as a $d$-martingale with zero-mean vector. Whenever the proce…
We study the asymptotic behavior of the sums of divisors when the integers are modelled with the Bernoulli random walk; We prealably study the correlation properties of the corresponding system.
The boundary problem is considered for inhomogeneous increasing random walks on the square lattice ${\mathbb Z}_+^2$ with weighted edges. Explicit solutions are given for some instances related to the classical and generalized number triangles.
The Bernoulli sieve is a random allocation scheme obtained by placing independent points with the uniform [0,1] law into the intervals made up by successive positions of a multiplicative random walk with factors taking values in the interval (0,1). Assuming that the number of points is equal to n we investigate the weak convergence, as n tends to infinity, of finite-dimensional distributions of the number of empty intervals within the occupancy range. A new argument enables us to relax the constraints imposed in previous papers on the distribution of the factor of the multiplicative random wal…
We propose an approach to construct Bernoulli trials $\{X_i, i\ge 1\}$ combining dependence and independence periods, and call it Bernoulli sequence with random dependence (BSRD). The structure of dependence, on the past $S_i = X_1 + \ldots + X_i$, {defines} a class of non-Markovian random walks of recent interest in the literature. In this paper, the dependence is activated by an auxiliary collection of Bernoulli trials $\{Y_i, i\ge 1\}$, called {\it memory switch sequence}. We introduce the concept of {\it memory lapses property}, which {is} characterized by intervals of consecutive independ…
We consider the $N$-particle noncolliding Bernoulli random walk --- a discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of $N$ independent simple random walks with steps $\in\{0,1\}$ by conditioning that they never collide. We study the asymptotic behavior of local statistics of this process started from an arbitrary initial configuration on short times $T\ll N$ as $N\to+\infty$. We show that if the particle density of the initial configuration is bounded away from $0$ and $1$ down to scales $\mathsf{D}\ll T$ in a neighborhood of size $\mathsf{Q}\gg T$ of some locatio…
We consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of trees. A sufficient condition is established for Galton-Watson trees.
We present a version of the RSK correspondence based on the Pitman transform and geometric considerations. This version unifies ordinary RSK, dual RSK and continuous RSK. We show that this version is both a bijection and an isometry, two crucial properties for taking limits of last passage percolation models. We use the bijective property to give a non-computational proof that dual RSK maps Bernoulli walks to nonintersecting Bernoulli walks.
Probabilistic models of random walks in random sceneries give rise to examples of probability-preserving dynamical systems. A point in the state spaces consists of a walk-trajectory and a scenery, and its `motion' corresponds to shifting the time-origin. These models were proposed as natural examples of non-Bernoulli K-automorphisms by Adler, Ornstein and Weiss. This was proved in a famous analysis by Kalikow using Ornstein's Very Weak Bernoulli characterization of Bernoulli processes. Since then, various authors have generalized this construction to give other examples, including some smooth …
We show that the Bernoulli part extraction method can be used to obtain approximate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term, that is with explicit parameters and universal constants. We also show that our estimates allow to recover Gnedenko and Gamkrelidze local limit theorems. We further establish by this method a local limit theorem with effective remainder for random walks in random scenery.
Let Z_N be the number of self-avoiding paths of length N starting from the origin on the infinite cluster obtained after performing Bernoulli percolation on Z^d with parameter p>p_c(Z^d). The object of this paper is to study the connective constant of the dilute lattice \limsup_{N\to \infty} Z_N^{1/N}, which is a non-random quantity. We want to investigate if the inequality \limsup_{N\to \infty} (Z_N)^{1/N} \le \lim_{N\to \infty} E[Z_N]^{1/N} obtained with the Borel-Cantelli Lemma is strict or not. In other words, we want to know the the quenched and annealed versions of the connective constan…
In this paper we study height fluctuations of random lozenge tilings of polygonal domains on the triangular lattice through nonintersecting Bernoulli random walks. For a large class of polygons which have exactly one horizontal upper boundary edge, we show that these random height functions converge to a Gaussian Free Field as predicted by Kenyon and Okounkov [28]. A key ingredient of our proof is a dynamical version of the discrete loop equations as introduced by Borodin, Guionnet and Gorin [5], which might be of independent interest.
We show that almost any one-dimensional projection of a suitably scaled random walk on a hypercube, inscribed in a hypersphere, converges weakly to an Ornstein-Uhlenbeck process as the dimension of the sphere tends to infinity. We also observe that the same result holds when the random walk is replaced with spherical Brownian motion. This latter result can be viewed as a "functional" generalisation of Poincaré's observation for projections of uniform measure on high dimensional spheres; the former result is an analogous generalisation of the Bernoulli-Laplace central limit theorem. Given the r…
We prove strong invariance principle between a transient Bessel process and a certain nearest neighbor (NN) random walk that is constructed from the former by using stopping times. It is also shown that their local times are close enough to share the same strong limit theorems. It is shown furthermore, that if the difference between the distributions of two NN random walks are small, then the walks themselves can be constructed so that they are close enough. Finally, some consequences concerning strong limit theorems are discussed.
We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and Häggström and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z}^d$, namely, for some critical value $λ_{\mathrm{c}} >0$ of the bias, it holds that the asymptotic linear speed $\overline{\mathrm{v}}$ of the walk is strictly positive if the bias $λ$ is strictly smaller than $λ_{\mathrm{c}}$, whereas $\overline{\mathrm{v}}=0$ if $λ\geq λ_{\mathrm{c}}$. We show that at the critical bias $λ= λ_{\mat…
A probabilistic construction for the solution of a general class of high order heat-type equations is constructed in terms of the scaling limit of random walks in the complex plane.
Given a $1$-cocycle $b$ with coefficients in an orthogonal representation, we show that any finite dimensional summand of $b$ is cohomologically trivial if and only if $\| b(X_n) \|^2/n$ tends to a constant in probability, where $X_n$ is the trajectory of the random walk $(G,μ)$. As a corollary, we obtain sufficient conditions for $G$ to satisfy Shalom's property $H_{\mathrm{FD}}$. Another application is a convergence to a constant in probability of $μ^{*n}(e) -μ^{*n}(g)$, $n\gg m$, normalized by its average with respect to $μ^{*m}$, for any finitely generated amenable group without infinite v…
It is well known that adding "long edges (shortcuts)" to a regularly constructed graph will make the resulted model a small world. Recently, \cite{W} indicated that, among all long edges, those edges with length proportional to the diameter of the regularly constructed graph may play the key role. In this paper, we modify the original Newman-Watts small world by adding only long special edges to the $d$-dimensional lattice torus (with size $n^d$) according to \cite{W}, and show that the diameter of the modified model and the mixing time of random walk on it grow polynomially fast in $\ln n$.
We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $K \subset \mathbb{R}^d$ is an attractor of a finite iterated function system of $n\geq 2$ maps of the form $x \mapsto D^{-r_i} x + t_i \ (i=1, \ldots, n)$, where $D$ is an expanding $d\times d$ integer matrix, and is the same for all the maps, and $r_{i} \in\mathbb{N}$, under an irrationality condition on the translation parts $t_i$, almost every p…
We show that the `erasing-larger-loops-first' (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpinski gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpinski gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the `standard' self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The e…
