Our model is a constrained homogeneous random walk in a nonnegative orthant Z_+^d. The convergence to stationarity for such a random walk can often be checked by constructing a Lyapunov function. The same Lyapunov function can also be used for computing approximately the stationary distribution of this random walk, using methods developed by Meyn and Tweedie. In this paper we show that, for this type of random walks, computing the stationary probability exactly is an undecidable problem: no algorithm can exist to achieve this task. We then prove that computing large deviation rates for this model is also an undecidable problem. We extend these results to a certain type of queueing systems. The implication of these results is that no useful formulas for computing stationary probabilities and large deviations rates can exist in these systems.
We extend the notion of the associated random walk and the Wald martingale in random walks where the increments are independent and identically distributed to the more general case of stationary ergodic increments. Examples are given where the increments are Markovian or Gaussian, and an application in queueing is considered.
In this paper, we consider the state-dependent reflecting random walk on a half-strip. We provide explicit criteria for (positive) recurrence, and an explicit expression for the stationary distribution. As a consequence, the light-tailed behavior of the stationary distribution is proved under appropriate conditions. The key idea of the method employed here is the decomposition of the trajectory of the random walk and the main tool is the intrinsic branching structure buried in the random walk on a strip, which is different from the matrix-analytic method.
In this paper, we consider the $(1,R)$ state-dependent reflecting random walk (RW) on the half line, allowing the size of jumps to the right at maximal $R$ and to the left only 1. We provide an explicit criterion for positive recurrence and the explicit expression of the stationary distribution based on the intrinsic branching structure within the walk. As an application, we obtain the tail asymptotic of the stationary distribution in the "near critical" situation.
We introduce a variation of strong stationary times for random walks on the symmetric group. Rather than proceed in the usual fashion of accumulating larger and larger blocks of cards which may be in any order, we wait for pairs of cards to `interact', and bound the mixing time by the time taken for enough interactions to occur.
We construct a two-dimensional counterexample of a random walk in random environment (RWRE). The environment is stationary, mixing and perturbative, and the corresponding RWRE has non-trivial probability to wander off to the upper right. This is in contrast to the 0-1-law that holds for i.i.d.\ environments.
In this paper, we consider the $(L,1)$ state-dependent reflecting random walk (RW) on the half line, which is a RW allowing jumps to the left at a maxial size $L$. For this model, we provide an explicit criterion for (positive) recurrence and an explicit expression for the stationary distribution.As an application, we prove the geometric tail asymptotic behavior of the stationary distribution under certain conditions. The main tool employed in the paper is the intrinsic branching structure within the $(L,1)$-random walk.
In [4], we examined the use of coupling to obtain bounds on the mixing time of statistics on Markov chains. In the present paper, we consider the same general problem, but using strong stationary times rather than coupling. We discuss various types of behaviour that may occur when this is attempted, and analyse a variety of examples.
Let $Γ$ act on a countable set V with only finitely many orbits. Given a $Γ$-invariant random environment for a Markov chain on V and a random scenery, we exhibit, under certain conditions, an equivalent stationary measure for the environment and scenery from the viewpoint of the random walker. Such theorems have been very useful in investigations of percolation on quasi-transitive graphs.
We consider a two dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction, which are referred to as upward jumps. We are interested in the tail asymptotic behavior of its stationary distribution, provided it exists. Assuming the upward jump size distributions have light tails, we completely find the rough tail asymptotics of the marginal stationary distributions in all directions. This generalizes the corresponding results for the …
In this work, we focus on the stationary analysis of a specific class of continuous time Markov-modulated reflected random walks in the quarter plane with applications in the modelling of two-node Markov-modulated queueing networks with coupled queues. The transition rates of the two-dimensional process depend on the state of a finite state Markovian background process. Such a modulation is space homogeneous in the set of inner states of the two-dimensional lattice but may be different in the set of states at its boundaries. To obtain the stationary distribution, we apply the power series appr…
We construct strong stationary dual chains for Ising model on a circle, non-symmetric random walk on square lattice and a random walk on hypercube. The strong stationary dual chains are all sharp and have the same state space as original chains. We use Möbius monotonicity of these chains with respect to natural orderings of the corresponding state spaces. This method provides an easy way to find eigenvalues in the Ising model and for a random walk on hypercube. Keywords: Markov chains; stochastic monotonicity; eigenvalues; Möbius monotonicity; strong stationary duality; strong stationary times…
We prove a formula for the speed of distance stationary random sequences. A particular case is the classical formula for the largest Lyapunov exponent of an i.i.d. product of two by two matrices in terms of a stationary measure on projective space. We apply this result to Poisson-Delaunay random walks on Riemannian symmetric spaces. In particular, we obtain sharp estimates for the asymptotic behavior of the speed of hyperbolic Poisson-Delaunay random walks when the intensity of the Poisson point process goes to zero. This allows us to prove that a dimension drop phenomena occurs for the harmon…
We survey distributional properties of $\mathbb{R}^d$-valued cocycles of finite measure preserving ergodic transformations (or, equivalently, of stationary random walks in $\mathbb{R}^d$) which determine recurrence or transience.
We consider a two dimensional reflecting random walk on the nonnegative integer quadrant. It is assumed that this reflecting random walk has skip free transitions. We are concerned with its time reversed process assuming that the stationary distribution exists. In general, the time reversed process may not be a reflecting random walk. In this paper, we derive necessary and sufficient conditions for the time reversed process also to be a reflecting random walk. These conditions are different from but closely related to the product form of the stationary distribution.
We examine the stationary distribution of random walks on directed graphs. In particular, we focus on the {\em principal ratio}, which is the ratio of maximum to minimum values of vertices in the stationary distribution. We give an upper bound for this ratio over all strongly connected graphs on $n$ vertices. We characterize all graphs achieving the upper bound and we give explicit constructions for these extremal graphs. Additionally, we show that under certain conditions, the principal ratio is tightly bounded. We also provide counterexamples to show the principal ratio cannot be tightly bou…
We consider random walks in a random environment of the type p_0+γξ_z, where p_0 denotes the transition probabilities of a stationary random walk on \BbbZ^d, to nearest neighbors, and ξ_z is an i.i.d. random perturbation. We give an explicit expansion, for small γ, of the asymptotic speed of the random walk under the annealed law, up to order 2. As an application, we construct, in dimension d\ge2, a walk which goes faster than the stationary walk under the mean environment.
This paper is a sequel to Chaika and Krishnan [arXiv:1612.00434]. We again consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice Z^d. We assume that once walks meet, they coalesce. We consider various entropic properties of these systems. We show that in systems with completely positive entropy, bi-infinite trajectories must carry entropy. In the case of directed walks in dimension 2 we show that positive entropy guarantees that all trajectories cannot be bi-infinite. To show that our theorems are proper, we construct a stationary di…
For a Markov chain $Y$ with values in a Polish space, consider the entrance Markov chain obtained by sampling $Y$ at the moments when it enters a fixed set $A$ from its complement $A^c$. Similarly, consider the exit Markov chain, obtained by sampling $Y$ at the exit times from $A^c$ to $A$. This paper provides a framework for analysing invariant measures of these two types of Markov chains in the case when the initial chain $Y$ has a known $σ$-finite invariant measure. Under certain recurrence-type assumptions ($Y$ can be transient), we give explicit formulas for invariant measures of these ch…
We focus on the three-state quantum walk(QW) in one dimension. In this paper, we give the stationary measure in general condition, originated from the eigenvalue problem. Firstly, we get the transfer matrices by our new recipe, and solve the eigenvalue problem. Then we obtain the general form of the stationary measure for concrete initial state and eigenvalue. We also show some specific examples of the stationary measure for the three-state QW. One of the interesting and crucial future problems is to make clear the whole picture of the set of stationary measures.
This paper studies Markov chains on the chambers of real hyperplane arrangements, a model that generalizes famous examples, such as the Tsetlin library and riffle shuffles. We discuss cutoff for the Tsetlin library for general weights, and we give an exact formula for the separation distance for the hyperplane arrangement walk. We introduce lower bounds, which allow for the first time to study cutoff for hyperplane arrangement walks under certain conditions. Using similar techniques, we also prove a uniform lower bound for the mixing time of Glauber dynamics on a monotone system.
We propose a class of models of random walks in a random environment where an exact solution can be given for a stationary distribution. The tool is the detailed balance equations.
We consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice. We assume that once walks meet, they coalesce. In $2d$, we classify the collective behavior of these walks under mild assumptions: they either coalesce almost surely or form bi-infinite trajectories. Bi-infinite trajectories form measure-preserving dynamical systems, have a common asymptotic direction in $2d$, and possess other nice properties. We use our theory to classify the behavior of compatible families of semi-infinite geodesics in stationary first- and last-passage per…
We define a random walk on the set of primitive points of $\mathbb{Z}^d$. We prove that for walks generated by measures satisfying mild conditions these walks are recurrent in a strong sense. That is, we show that the associated Markov chains are positive recurrent and there exists a unique stationary measure for the random walk.
We study random walk with unbounded jumps in random environment. The environment is stationary and ergodic, uniformly elliptic and decays polynomially with speed $Dj^{-(3+\varepsilon_0)}$ for some small $\varepsilon_0>0$ and proper $D>0.$ We prove a law of large number with positive velocity under the condition that the annealed mean of the hitting time of the positive half lattice is finite. Secondly, we consider birth and death process with bounded jumps in stationary and ergodic environment. Under the uniformly elliptic condition, we prove a law of large number and give the explicit formula…
We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 times 2 unitary matrix U. In our previous paper [15], we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except U is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving …
We consider a general multidimensional affine recursion with corresponding Markov operator $P$ and a unique $P$-stationary measure. We show spectral gap properties on Hölder spaces for the corresponding Fourier operators and we deduce convergence to stable laws for the Birkhoff sums along the recursion. The parameters of the stable laws are expressed in terms of basic quantities depending essentially on the matricial multiplicative part of $P$. Spectral gap properties of $P$ and homogeneity at infinity of the $P$-stationary measure play an important role in the proofs.
We study the analogues of irreducibility, period, and communicating classes for open quantum random walks, as defined by Attal et al. (J. Stat. Phys., 2012). We recover results similar to the standard ones for Markov chains, in terms of ergodic behavior, decomposition into irreducible subsystems, and characterization of stationary states.
We consider random walks in dynamic random environments given by Markovian dynamics on $\mathbb{Z}^d$. We assume that the environment has a stationary distribution $μ$ and satisfies the Poincaré inequality w.r.t. $μ$. The random walk is a perturbation of another random walk (called "unperturbed"). We assume that also the environment viewed from the unperturbed random walk has stationary distribution $μ$. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for…
We prove central limit theorem under diffusive scaling for the displacement of a random walk on ${\mathbb Z}^d$ in stationary divergence-free random drift field, under the ${\mathcal H}_{-1}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor be stationary and square integrable. This improves the best existing result of Komorowski, Landim and Olla (2012), where it is assumed that the stream tensor be in ${\mathcal L}^{\max\{2+δ,d\}}$, with $δ>0$. Our proof relies on the relaxed sector condition of Horváth, Tóth and Vető (2012), and is technical…
We study dynamic random conductance models on $\mathbb{Z}^2$ in which the environment evolves as a reversible Markov process that is stationary under space-time shifts. We prove under a second moment assumption that two conditionally independent random walks in the same environment collide infinitely often almost surely. These results apply in particular to random walks on dynamical percolation.
A continuous-time random walk in the quarter plane with homogeneous transition rates is considered. Given a non-negative reward function on the state space, we are interested in the expected stationary performance. Since a direct derivation of the stationary probability distribution is not available in general, the performance is approximated by a perturbed random walk, whose transition rates on the boundaries are changed such that its stationary probability distribution is known in closed form. A perturbed random walk for which the stationary distribution is a sum of geometric terms is consid…
In this paper, we present a numerical framework for constructing bounds on stationary performance measures of random walks in the positive orthant using the Markov reward approach. These bounds are established in terms of stationary performance measures of a perturbed random walk whose stationary distribution is known explicitly. We consider random walks in an arbitrary number of dimensions and with a transition probability structure that is defined on an arbitrary partition of the positive orthant. Within each component of this partition the transition probabilities are homogeneous. This enab…
Let $(G,ρ)$ be a stationary random graph, and use $B^G_ρ(r)$ to denote the ball of radius $r$ about $ρ$ in $G$. Suppose that $(G,ρ)$ has annealed polynomial growth, in the sense that $\mathbb{E}[|B^G_ρ(r)|] \leq O(r^k)$ for some $k > 0$ and every $r \geq 1$. Then there is an infinite sequence of times $\{t_n\}$ at which the random walk $\{X_t\}$ on $(G,ρ)$ is at most diffusive: Almost surely (over the choice of $(G,ρ)$), there is a number $C > 0$ such that \[ \mathbb{E} \left[\mathrm{dist}_G(X_0, X_{t_n})^2 \mid X_0 = ρ, (G,ρ)\right]\leq C t_n\qquad \forall n \geq 1\,. \] This result is new ev…
We propose two numerical schemes for approximating quasi-stationary distributions (QSD) of finite state Markov chains with absorbing states. Both schemes are described in terms of certain interacting chains in which the interaction is given in terms of the total time occupation measure of all particles in the system and has the impact of reinforcing transitions, in an appropriate fashion, to states where the collection of particles has spent more time. The schemes can be viewed as combining the key features of the two basic simulation-based methods for approximating QSD originating from the wo…
The probability that a one dimensional excited random walk in stationary ergodic and elliptic cookie environment is transient to the right (left) is either zero or one. This solves a problem posed by Kosygina and Zerner [8].
We construct forests that span $\mathbb{Z}^d$, $d\geq2$, that are stationary and directed, and whose trees are infinite, but for which the subtrees attached to each vertex are as short as possible. For $d\geq3$, two independent copies of such forests, pointing in opposite directions, can be pruned so as to become disjoint. From this, we construct in $d\geq3$ a stationary, polynomially mixing and uniformly elliptic environment of nearest-neighbor transition probabilities on $\mathbb{Z}^d$, for which the corresponding random walk disobeys a certain zero--one law for directional transience.
We provide a random walk in random scenery representation of a new class of stable self-similar processes with stationary increments introduced recently by Jung, Owada and Samorodnitsky. In the functional limit theorem they provided, only a single instance of this class arose as a limit. We construct a model in which a significant portion of processes in this new class is obtained as a limit.
We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order n^2 steps are necessary and suffice for convergence to the stationary distribution.
The sequence of the simple random walks on Hamming schemes $\{H(n, q)\}_{n=1}^{\infty}$ has a cutoff phenomenon for each integer $q$ greater than or equal to $3$. In this paper, for the sequence of simple random walks on Hamming schemes $\{H(n, q)\}_{n=1}^{\infty}$ with $q\geq 3$, we give a simple majorant and a sharp minorant function for total variance distances between transition distributions and stationary distributions.
We prove a formula for the speed of distance stationary random sequences generalizing the law of large numbers of Karlsson and Ledrappier. A particular case is the classical formula for the largest Lyapunov exponent of i.i.d.\ matrix products, but our result has applications in various different contexts. In many situations it gives a method to estimate the speed, and in others it allows to obtain results of dimension drop for escape measures related to random walks. We show applications to stationary reversible random trees with conductances, Bernoulli bond percolation of Cayley graphs, and r…
We use coupling to study the time taken until the distribution of a statistic on a Markov chain is close to its stationary distribution. Coupling is a common technique used to obtain upper bounds on mixing times of Markov chains, and we explore how this technique may be used to obtain bounds on the mixing of a statistic instead.
We consider Activated Random Walks on arbitrary finite networks, with particles being inserted at random and absorbed at the boundary. Despite the non-reversibility of the dynamics and the lack of knowledge on the stationary distribution, we explicitly determine the relaxation time of the process, and prove that separation cutoff is equivalent to the product condition. We also provide sharp estimates on the center and width of the cutoff window. Finally, we illustrate those results by establishing explicit separation cutoffs on various networks, including: (i) large finite subgraphs of any fix…
Random walk on $\mathbb{N}$ with negative drift and absorption at 0, when conditioned on survival, has uncountably many invariant measures (quasi-stationary distributions, qsd) $ν_c$. We study a Fleming-Viot(FV) particle system driven by this process and show that mean normalized densities of the FV unique stationary measure converge to the minimal qsd, $ν_0$, as $N \to \infty$. Furthermore, every other qsd of the random walk ($ν_c$, $c>0$) corresponds to a metastable state of the FV particle system.
