On some random walks on Z in random medium

On the recurrence of random walks on [formula omitted] in random medium

Behavior of random walks on [formula omitted] in Gibbsian medium

This monograph focuses on random walks in the quarter-plane. Such random walks are frequently used to model queueing systems and the invariant measure of a random walk is of major importance in studying the performance of these systems. In special cases the invariant measure of a random walk can be expressed as a geometric product-form and performance measures can readily be obtained. In general, however, no tractable closed-form expressions are available for the invariant measure and exact performance measures are not readily obtained. The aim of this monograph is two-fold. On one hand we consider measures that are a sum of geometric terms. We characterize the random walks in the quarter-plane of which the invariant measure is of this form. This extends the class of random walks for which tractable closed-form results can be obtained. On the other hand we develop approximation schemes that provide analytical upper and lower bounds on performance for the case that no tractable closed-form expressions for the invariant measure are available. It is shown that the necessary conditions that are required to have an invariant measure which is a sum of geometric terms are as follows: Each geometric term must individually satisfy the balance equations in the interior of the state space and the geometric terms in an invariant measure must have a pairwise-coupled structure. Moreover, at least one of the coecients in the linear combination must be negative. When the invariant measure of the random walk is a sum of innitely many geometric terms, it is also required that the random walk does not have transitions to the North, Northeast or East in the interior of the state space. Based on the necessary conditions described above, we develop an algorithm which determines whether the invariant measure of a given random walk is a sum of geometric terms. For the case that the invariant measure of this random walk is not a sum of geometric terms, we have developed an approximation scheme to find upper and lower bounds for the performance measures. This approximation scheme is based on a Markov reward approach to error bounds. The bounds determined by our approximation scheme are established in terms of a perturbed random walk of which the invariant measure is a sum of geometric terms. Numerical results reveal that using a perturbed random walk of which the invariant measure is a sum of multiple geometric terms may lead to better bounds than using perturbed random walks of which the invariant measure is of product-form. Finally, we have also extended this approximation scheme to two-dimensional finite random walks.

An optical system consisting of a laser source and two independent consecutive phase-only spatial light modulators (SLMs) is shown to accurately simulate a generated random beam (first SLM) after interaction with a stationary random medium (second SLM). To illustrate the range of possibilities, a recently introduced class of random optical frames is examined on propagation in free space and several weak turbulent channels with Kolmogorov and non-Kolmogorov statistics.

Let $${\mathcal{T}}$$ be a rooted Galton–Watson tree with offspring distribution {p k } that has p 0 = 0, mean m = ∑ kp k > 1 and exponential tails. Consider the λ-biased random walk {X n } n ≥ 0 on $${\mathcal{T}}$$ ; this is the nearest neighbor random walk which, when at a vertex v with d v offspring, moves closer to the root with probability λ/(λ + d v ), and moves to each of the offspring with probability 1/(λ + d v ). It is known that this walk has an a.s. constant speed $${\tt v} = \lim_n |X_n|/n$$ (where |X n | is the distance of X n from the root), with $${\tt v} > 0$$ for 0 < λ < m and $${\tt v} = 0$$ for λ ≥ m. For all λ ≤ m, we prove a quenched CLT for $$|X_n| - n{\tt v}$$ . (For λ > m the walk is positive recurrent, and there is no CLT.) The most interesting case by far is λ = m, where the CLT has the following form: for almost every $${\mathcal{T}}$$ , the ratio $$|X_{[nt]}|/\sqrt{n}$$ converges in law as n → ∞ to a deterministic multiple of the absolute value of a Brownian motion. Our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle (previously, such a measure was explicitly known only for λ = 1) and the construction of appropriate harmonic coordinates.

Synthesis of automata, asymptotically optimal in stationary random media

Spontaneous collapse models use non-linear stochastic modifications of the Schrödinger equation to suppress superpositions of eigenstates of the measured observable and drive the state to an eigenstate. It was recently demonstrated that the Born rule for transition probabilities can be modeled using the linear Schrödinger equation with a Hamiltonian represented by a random matrix from the Gaussian unitary ensemble. The matrices representing the Hamiltonian at different time points throughout the observation period are assumed to be independent. Instead of suppressing superpositions, such Schrödinger evolution makes the state perform an isotropic random walk on the projective space of states. The relative frequency of reaching different eigenstates of an arbitrary observable in the random walk is shown to satisfy the Born rule. Here, we apply this methodology to investigate the behavior of a particle in the context of the double-slit experiment with measurement. Our analysis shows that, in this basic case, the evolution of the particle’s state can be effectively captured through a random walk on a two-dimensional submanifold of the state space. This random walk reproduces the Born rule for the probability of finding the particle near the slits, conditioned on its arrival at one of them. To ensure that this condition is satisfied, we introduce a drift term representing a change in the variance of the position observable for the state. It is argued that the drift term accounts for the energy transfer and trapping incurred during the particle’s interaction with the detector. A drift-free model, based on equivalence classes of states indistinguishable by the detector, is also considered. The resulting random walk, with or without drift, serves as a suitable model for describing the transition from the initial state to an eigenstate of the measured observable in the experiment, offering new insights into its potential underlying mechanisms.

Let $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $ -approximable points and for the set of inhomogeneous multiplicative $\psi $ -approximable points on nondegenerate planar curves. Our results extend theorems of Huang [‘Hausdorff theory of dual approximation on planar curves’, J. reine angew. Math.740 (2018), 63–76] and Beresnevich and Velani [‘A note on three problems in metric Diophantine approximation’, in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211–229] from s-Hausdorff measure, where $s\in \mathbb R$ , to the more general g-Hausdorff measure, where g is a suitable class of dimension functions.

Let $\\omega = (p_x)_{x\\in\\mathbb{Z}}$ be an i.i.d. collection of (0, 1)-valued random variables. Given $\\omega$, let $(X_n)_{n \\geq 0}$ be the Markov chain on $\\mathbb{Z}$ defined by $X_0 = 0$ and $X_{n + 1} = X_n + 1(\\operatorname{resp}. X_n - 1)$ with probability $p_{X_n}(\\operatorname{resp}.1 - p_{X_n})$. It is shown that $X_n/n$ satisfies a large deviation principle with a continuous rate function, that is, $\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\log P_\\omega(X_n = \\lfloor\\theta_nn\\rfloor) = -I(\\theta) \\omega-\\mathrm{a.s.}\\text{for} \\theta_n\\rightarrow\\in\\lbrack -1, 1\\rbrack.$ First, we derive a representation of the rate function $I$ in terms of a variational problem. Second, we solve the latter explicitly in terms of random continued fractions. This leads to a classification and qualitative description of the shape of $I$. In the recurrent case $I$ is nonanalytic at $\\theta = 0$. In the transient case $I$ is nonanalytic at $\\theta = -\\theta_c, 0, \\theta_c$ for some $\\theta_c \\geq 0$, with linear pieces in between.

Course 4 Random matrices, the ulam problem, directed polymers & growth models, and sequence matching

We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary. Physical motivation for the process originates with ideal gas models in the Knudsen regime, with particles reflecting off microscopically rough surfaces. We classify the process into recurrent and transient cases. We also give almost-sure results on the long-term behaviour of the location of the particle, including a super-diffusive rate of escape in the transient case. A key step in obtaining our results is to relate our process to an instance of a one-dimensional stochastic process with asymptotically zero drift, for which we prove some new almost-sure bounds of independent interest. We obtain some of these bounds via an application of general semimartingale criteria, also of some independent interest.

We consider infinite Galton-Watson trees without leaves together with i.i.d.~random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, to build explicit invariant measures. We apply this result, together with the ergodic theory on Galton-Watson trees developed in \cite{LPP95}, to the computation of Hausdorff dimensions of harmonic measures in two cases. The first one is the harmonic measure of the (transient) $λ$-biased random walk on Galton-Watson trees, for which the invariant measure and the dimension were not explicitly known. The second case is a model of random walk on a Galton-Watson trees with random lengths for which we compute the dimensions of the harmonic measure and show dimension drop phenomenon for the natural metric on the boundary and another metric that depends on the random lengths.

We extend results of Y. Benoist and J.-F. Quint concerning random walks on homogeneous spaces of simple Lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily Zariski dense, but satisfies some expansion properties for the adjoint action. Using these dynamical results, we study Diophantine properties of typical points on some self-similar fractals in $${\mathbb {R}}^d$$ . As examples, we show that for any self-similar fractal $$\mathcal {K}\subseteq {\mathbb {R}}^d$$ satisfying the open set condition (for instance any translate or dilate of Cantor’s middle thirds set or of a Koch snowflake), almost every point with respect to the natural measure on $$\mathcal {K}$$ is not badly approximable. Furthermore, almost every point on the fractal is of generic type, which means (in the one-dimensional case) that its continued fraction expansion contains all finite words with the frequencies predicted by the Gauss measure. We prove analogous results for matrix approximation, and for the case of fractals defined by Mobius transformations.

Majoration du noyau de la chaleur en environnement stationnaire de conductances

A lock step walk is a one-dimensional integer lattice walk in discrete time. Suppose that initially there are infinitely many walkers on the nonnegative even integer sites. At each moment of time, every walker moves either to its left or to its right with equal probability. The only constraint is that no two walkers can occupy the same site at the same time. Hence we describe the walk as vicious. It is proved that as time tends to infinity, a certain limiting conditional distribution of the displacement of the leftmost walker is identical to the limiting distribution of the (scaled) largest eigenvalue of a random GOE matrix (GOE Tracy-Widom distribution). The proof is based on the bijection between path configurations and semistandard Young tableaux established recently by Guttmann, Owczarek, and Viennot. The distribution of semistandard Young tableaux is analyzed using the Hankel determinant expression for the probability obtained from the work of Rains and the author. The asymptotics of the Hankel determinant are then obtained by applying the Deift-Zhou steepest-descent method to the Riemann-Hilbert problem for the related orthogonal polynomials. © 2000 John Wiley & Sons, Inc.