We start by introducing avoidance coupling of Markov chains, with an overview of existing results. We then introduce and motivate a new notion, uniform avoidance coupling. We show that the only Markovian avoidance coupling on a cycle is of this type, and that uniform avoidance coupling of simple random walks is impossible on trees, and prove that it is possible on several classes of graphs. We also derive a condition on the vertex neighborhoods in a graph equivalent to that graph admitting a uniform avoidance coupling of simple random walks, and an algorithm that tests this with run time polynomial in the number of vertices.
A coupling of random walkers on the same finite graph, who take turns sequentially, is said to be an avoidance coupling if the walkers never collide. Previous studies of these processes have focused almost exclusively on complete graphs, in particular how many walkers an avoidance coupling can include. For other graphs, apart from special cases, it has been unsettled whether even two non-colliding simple random walkers can be coupled. In this article, we construct such a coupling on (i) any $d$-regular graph avoiding a fixed subgraph depending on $d$; and (ii) any square-free graph with minimu…
Answering a question by Angel, Holroyd, Martin, Wilson and Winkler, we show that the maximal number of non-colliding coupled simple random walks on the complete graph $K_N$, which take turns, moving one at a time, is monotone in $N$. We use this fact to couple $\lceil \frac N4 \rceil$ such walks on $K_N$, improving the previous $Ω(N/\log N)$ lower bound of Angel et al. We also introduce a new generalization of simple avoidance coupling which we call partially ordered simple avoidance coupling and provide a monotonicity result for this extension as well.
We examine the question of whether a collection of random walks on a graph can be coupled so that they never collide. In particular, we show that on the complete graph on n vertices, with or without loops, there is a Markovian coupling keeping apart Omega(n/log n) random walks, taking turns to move in discrete time.
We show that a coupling of non-colliding simple random walkers on the complete graph on $n$ vertices can include at most $n - \log n$ walkers. This improves the only previously known upper bound of $n-2$ due to Angel, Holroyd, Martin, Wilson, and Winkler ({\it Electron.~Commun.~Probab.~18}, 2013). The proof considers couplings of i.i.d.~sequences of Bernoulli random variables satisfying a similar avoidance property, for which there is separate interest. Our bound in this setting should be closer to optimal.
We give a survey and unified treatment of functional integral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak self-avoidance, and a model of walks and loops. Our representation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigorous renormalization group analyses of self-avoiding walk models in dimension 4. For the models without loops, the integral representations involve fermions, and we also provide an introduction to fermionic integra…
We prove that a uniformized variant of both the Rosenthal walk \cite{Rosenthal} and the Kac random walk \cite{Kac} on SO(n) mixes in $\cO(n^3)$ steps in total variation distance. The proof also extends easily to Rosenthal walk with fixed angle $θ\neq π$. To the best of our knowledge, this is the first polynomial time bound for both walks. The techniques employed are mainly from representation theory of SO(n). But a crucial new ingredient is the interpretation of the Fourier coefficients of the character ratio as counting the number of particle cascade paths arising from the classical branching…
We study simple random walk on the uniform spanning tree on Z^2 . We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean balls and balls in the intrinsic graph metric. In particular, we prove that the spectral dimension of the uniform spanning tree on Z^2 is 16/13 almost surely.
We study the entropy of the distribution of the set R_n of vertices visited by a simple random walk on a graph with bounded degrees in its first n steps. It is shown that this quantity grows linearly in the expected size of R_n if the graph is uniformly transient, and sublinearly in the expected size if the graph is uniformly recurrent with subexponential volume growth. This in particular answers a question asked by Benjamini, Kozma, Yadin and Yehudayoff (arXiv:0903.3179v1).
We study the 2-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the direction of a previously visited site. The uniform prudent walk has been investigated with combinatorial techniques in [Bousquet-Mélou, 2010], while another variant, the kinetic prudent walk has been analyzed in detail in [Beffara, Friedli and Velenik, 2010]. In this paper, we prove that the $2$-dimensional uniform prudent walk is ballistic and follows one of…
We prove results for random walks in dynamic random environments which do not require the strong uniform mixing assumptions present in the literature. We focus on the "environment seen from the walker"-process and in particular its invariant law. Under general conditions it exists and is mutually absolutely continuous to the environment law. With stronger assumptions we obtain for example uniform control on the density or a quenched CLT. The general conditions are made more explicit by looking at hidden Markov models or Markov chains as environment and by providing simple examples.
We study the pioneer points of the simple random walk on the uniform infinite planar quadrangulation (UIPQ) using an adaptation of the peeling procedure of Angel to the quadrangulation case. Our main result is that, up to polylogarithmic factors, $n^3$ pioneer points have been discovered before the walk exits the ball of radius $n$ in the UIPQ. As a result we verify the KPZ relation in the particular case of the pioneer exponent and prove that the walk is subdiffusive with exponent less than 1/3. Along the way, new geometric controls on the UIPQ are established.
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{uniform} invariance principle for the random walk. This means that the rescaled trajectory of length $n$ is (in a certain sense) close enough to the Brownian motion, uniformly with respect to the choice of the starting location in an interval of length $O(\sqrt{n})$ around the origin.
In this short note we give various near optimal characterizations of random walks over finite Abelian groups with large maximum discrepancy from the uniform measure. We also provide several interesting connections to existing results in the literature.
We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (i) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (ii) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third m…
We present two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin. One is a limit measure which is obtained from the probability distribution of the walk at a long-time limit, and the other is a convergence in distribution for the walker's position in a rescaled space by time. In addition, as an application of the walk, we obtain discrete uniform limit measures from the 3-state walk with a delocalized initial state.
In a \emph{rotor walk} the exits from each vertex follow a prescribed periodic sequence. On an infinite Eulerian graph embedded periodically in $\R^d$, we show that any simple rotor walk, regardless of rotor mechanism or initial rotor configuration, visits at least on the order of $t^{d/(d+1)}$ distinct sites in $t$ steps. We prove a shape theorem for the rotor walk on the comb graph with i.i.d.\ uniform initial rotors, showing that the range is of order $t^{2/3}$ and the asymptotic shape of the range is a diamond. Using a connection to the mirror model and critical percolation, we show that r…
In this paper we construct uniformly expanding random walks on smooth manifolds. In higher dimensions, our definition of uniform expansion measures the growth of subspaces rather than single vectors. Potrie showed that given any open set $U$ of $\text{Diff}^\infty_{vol}(\mathbb{T}^2)$, there exists an uniformly expanding random walk $μ$ supported on a finite subset of $U$. In this paper we extend those results to closed manifolds of any dimension, building on the work of Potrie and Chung to build a robust class of examples.
We calculate explicit speeds for random walks in uniform degenerate random environments. For certain non-uniform random environments, we calculate speeds that are non-monotone.
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{conditional} invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$. One of the main tools for proving these conditional limit la…
Let x and y be chosen uniformly in a graph G. We find the limiting distribution of the length of a loop-erased random walk from x to y on a large class of graphs that include the discrete torus in dimensions 5 and above. Moreover, on this family of graphs we show that a suitably normalized finite-dimensional scaling limit of the uniform spanning tree is a Brownian continuum random tree.
Let x and y be points chosen uniformly at random from $\Z_n^4$, the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order $n^2 (\log n)^{1/6}$, resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on $\Z_n^4$ is the Brownian continuum random tree of Aldous. Our proofs use the techniques developed by Peres and Revelle, who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d-dimensional discrete torus for…
For any graph having a suitable uniform Poincare inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks evolving via time varying, uniformly elliptic conductances, provided the vertex conductances (i.e. reversing measures), increase in time. Such transition density upper bounds apply for discrete time uniformly lazy walks, with the matching lower bounds holding once the parabolic Harnack inequality is proved.
We consider the range of random walks up to time n, R_n, on graphs satisfying a uniform condition. This condition is characterized by potential theory. Not only all vertex transitive graphs but also many non-regular graphs satisfy the condition. We show certain weak laws of R_n from above and below. We also show that there is a graph such that it satisfies the condition and a sequence of the mean of R_n/n fluctuates. By noting the construction of the graph, we see that under the condition, the weak laws are best in a sense.
In this paper we study random walks on dynamical random environments in $1 + 1$ dimensions. Assuming that the environment is invariant under space-time shifts and fulfills a mild mixing hypothesis, we establish a law of large numbers and a concentration inequality around the asymptotic speed. The mixing hypothesis imposes a polynomial decay rate of covariances on the environment with sufficiently high exponent but does not impose uniform mixing. Examples of environments for which our methods apply include the contact process and Markovian environments with a positive spectral gap, such as the …
We consider uniform spanning tree (UST) in topological polygons with $2N$ marked points on the boundary with alternating boundary conditions. In [LPW21], the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE$_8$. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE$_2$. The conclusion is a generalization of [HLW20,Theorem 1.6] where the authors derive the scaling limit of the LERW branch of UST when $N=2$. When $N=2$, the limiting law is SLE$_2(-1,-1; -1, -1)$. However, the limiting…
In this note, we prove an $L^p$ uniform approximation of the fractional Brownian motion with Hurst exponent $0 < H < \frac{1}{2}$ by means of a family of continuous-time random walks imbedded on a given Brownian motion. The approximation is constructed via a pathwise representation of the fractional Brownian motion in terms of a standard Brownian motion. For an arbitrary choice $ε_k$ for the size of the jumps of the family of random walks, the rate of convergence of the approximation scheme is $O(ε_k^{p(1-2λ)+ 2(δ-1)})$ whenever $\max\{0,1-\frac{pH}{2}\}< δ< 1$, $λ\in \big(\frac{1-H}{2}, \frac…
We study a random walk in random environment on the non-negative integers. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i) points endowed with probabilities drawn from a symmetric distribution with heavy tails at 0 and 1, and (ii) `fast points' with a fixed systematic drift. Without these fast points, the model is related to the diffusion in heavy-tailed (`stable') random potential studied by Schumacher and Singh; the fast points perturb that model. The two components compete to d…
Let S_i be a random walk with standard exponential increments. We call \sum_{i=1}^k S_i its k-step area. The random variable V = \inf_{k \ge 1} \frac{2}{k(k+1)} \sum_{i=1}^k S_i plays important role in the study of so-called one-dimensional sticky particles model. We find the distribution of V and prove that P(V > t) = \sqrt{1-t} exp(-t/2) for t in [0,1]. We also show that the variables \min_{1 \le k \le n} \frac{2n}{k(k+1)} \sum_{i=1}^k U_{i, n} converge in distribution to V. Here U_{i, n} are the order statistics of n i.i.d. random variables uniformly distributed on [0,1].
The uniform spanning forest measure ($\mathsf{USF}$) on a locally finite, infinite connected graph $G$ with conductance $c$ is defined as a weak limit of uniform spanning tree measure on finite subgraphs. Depending on the underlying graph and conductances, the corresponding $\mathsf{USF}$ is not necessarily concentrated on the set of spanning trees. Pemantle~\cite{PR1991} showed that on $\mathbb{Z}^d$, equipped with the unit conductance $ c=1$, $\mathsf{USF}$ is concentrated on spanning trees if and only if $d \leq 4$. In this work we study the $\mathsf{USF}$ associated with conductances induc…
In this thesis, we study the diffusive and ballistic behaviors of random walk in random environment (RWRE) in an integer lattice with dimension at least 2. Our contributions are in three directions: a conditional law of large numbers and regeneration structures for RWRE in Gibbsian environments, quenched invariance principles for balanced elliptic (but non uniformly elliptic) environments, and a proof of the Einstein relation for balanced iid uniformly elliptic environments.
The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of the intrinsic metrics, measures and embeddings of the subsequential limits in this space are obtained, with it being proved in particular that the Hausdorff dimension of any limit in its intrinsic metric is almost surely equal to $8/5$. In addition, the tightness result is applied to deduce that the annealed law of the simple random walk on the two-dimensio…
Let {X_n,n\geq0} be a Markov chain on a general state space X with transition probability P and stationary probability π. Suppose an additive component S_n takes values in the real line R and is adjoined to the chain such that {(X_n,S_n),n\geq0} is a Markov random walk. In this paper, we prove a uniform Markov renewal theorem with an estimate on the rate of convergence. This result is applied to boundary crossing problems for {(X_n,S_n),n\geq0}. To be more precise, for given b\geq0, define the stopping time τ=τ(b)=inf{n:S_n>b}. When a drift μof the random walk S_n is 0, we derive a one-term Ed…
Suppose that $\CG$ is a finite, connected graph and $X$ is a lazy random walk on $\CG$. The lamplighter chain $X^\diamond$ associated with $X$ is the random walk on the wreath product $\CG^\diamond = \Z_2 \wr \CG$, the graph whose vertices consist of pairs $(f,x)$ where $f$ is a labeling of the vertices of $\CG$ by elements of $\Z_2$ and $x$ is a vertex in $\CG$. There is an edge between $(f,x)$ and $(g,y)$ in $\CG^\diamond$ if and only if $x$ is adjacent to $y$ in $\CG$ and $f(z) = g(z)$ for all $z \neq x,y$. In each step, $X^\diamond$ moves from a configuration $(f,x)$ by updating $x$ to $y$…
We consider the biased random walk on a tree constructed from the set of finite self-avoiding walks on a lattice, and use it to construct probability measures on infinite self-avoiding walks. The limit measure (if it exists) obtained when the bias converges to its critical value is conjectured to coincide with the weak limit of the uniform SAW. Along the way, we obtain a criterion for the continuity of the escape probability of a biased random walk on a tree as a function of the bias, and show that the collection of escape probability functions for spherically symmetric trees of bounded degree…
Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^3$, whose probability law is denoted by $\mathbf{P}$. For $\mathbf{P}$-a.s. realization of $\mathcal{U}$, the recurrence of the the simple random walk on $\mathcal{U}$ is proved in [5] and it is also demonstrated in [8] that two independent simple random walks on $\mathcal{U}$ collide infinitely often. In this article, we will give a quantitative estimate on the number of collisions of two independent simple random walks on $\mathcal{U}$, which provides another proof of the infinite collision property of $\mathcal{U}$.
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 …
Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. Peköz, Röllin, and Ross (2016, arXiv:1309.4183 [math.PR]) exploited characterizing distributional fixed point equations to obtain uniform error bounds for generalized gamma approximations using Stein's method. Here we show how monotone couplings arising with these fixed point equations can be used to obtain sharper tail bounds that, in many cases, outperform competing moment-based bounds and the uniform bounds obtainable with Stein's method. Applications are given to…
We consider time-dependent random walks among time-dependent conductances. For discrete time random walks, we show that, unlike the time-independent case, two-sided Gaussian heat kernel estimates are not stable under perturbations. This is proved by giving an example of a ballistic and transient time-dependent random walk on Z among uniformly elliptic time-dependent conductances. For continuous time random walks, we show the instability when the holding times are i.i.d. exp(1), and in contrast, we prove the stability when the holding times change by sites in such a way that the base measure is…
We study the random walk on the symmetric group $S_n$ generated by the conjugacy class of cycles of length $k$. We show that the convergence to uniform measure of this walk has a cut-off in total variation distance after $\frac{n}{k} log n$ steps, uniformly in $k = o(n)$ as $n \to \infty$. The analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles.
We bound the rate of convergence to uniformity for certain random walks on the complete monomial groups G \wr S_n for any group G. These results provide rates of convergence for random walks on a number of groups of interest: the hyperoctahedral group Z_2 \wr S_n, the generalized symmetric group Z_m \wr S_n, and S_m \wr S_n. These results provide benchmarks to which many other random walks, modeling a wide range of phenomena, may be compared using the comparison technique, thereby yielding bounds on the rates of convergence to uniformity for previously intractable random walks.
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…
A simple way to sample a uniform triangulation of the sphere with a fixed number $n$ of vertices is a Monte-Carlo method: we start from an arbitrary triangulation and flip repeatedly a uniformly chosen edge. We give a lower bound in $n^{5/4}$ on the mixing time of this Markov chain.
By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate 1 the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and prove that both the total variation distance and the separation distance to the uniform distribution exhibit a cutoff phenomenon, with mixing times that differ by a factor $2$. The results are extended to the family of log-concave d…
We show that the measure on markings of $\mathbf {Z}_n^d$, $d\geq3$, with elements of ${0,1}$ given by i.i.d. fair coin flips on the range $\mathcal {R}$ of a random walk $X$ run until time $T$ and 0 otherwise becomes indistinguishable from the uniform measure on such markings at the threshold $T=1/2T_{\mathrm {cov}}(\mathbf {Z}_n^d)$. As a consequence of our methods, we show that the total variation mixing time of the random walk on the lamplighter graph $\mathbf {Z}_2\wr \mathbf {Z}_n^d$, $d\geq3$, has a cutoff with threshold $1/2T_{\mathrm {cov}}(\mathbf {Z}_n^d)$. We give a general criteri…
We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by $3/β$, where $β\approx 1.624\dots$ is the growth exponent of three-dimens…
We prove a conjecture of Toth and Veto about the weak convergence of the self repelling random walk with directed edges under diffusive scaling to a uniform distribution.
Let $ p_n(x)=\int_0^\infty J_0(xt)[J_0(t)]^n xt\,\mathrm{d}\, t$ be Kluyver's probability density for $n$-step uniform random walks in the Euclidean plane. Through connection to a similar problem in 2-dimensional quantum field theory, we evaluate the third-order derivative $ p_5'''(0^{+})$ in closed form, thereby giving a new proof for a conjecture of J. M. Borwein. By further analogies to Feynman diagrams in quantum field theory, we demonstrate that $ p_n(x),0\leq x\leq 1$ admits a uniformly convergent Maclaurin expansion for all odd integers $ n\geq5$, thus settling another conjecture of Bor…
For a simple symmetric random walk in dimension $d\geq 3$, a uniform strong law of large numbers is proved for the number of sites with given local time up to time $n$.
This paper presents necessary and sufficient conditions for on- and off-diagonal transition probability estimates for random walks on weighted graphs. On the integer lattice and on may fractal type graphs both the volume of a ball and the mean exit time from a ball is independent of the centre, uniform in space. Here the upper estimate is given without such restriction and two-sided estimate is given if uniformity in the space assumed only for the mean exit time.
We give refined estimates for the discrete time and continuous time versions of some basic random walks on the symmetric and alternating groups $S_n$ and $A_n$. We consider the following models: random transposition, transpose top with random, random insertion, and walks generated by the uniform measure on a conjugacy class. In the case of random walks on $S_n$ and $A_n$ generated by the uniform measure on a conjugacy class, we show that in continuous time the $\ell^2$-cuttoff has a lower bound of $(n/2)\log n$. This result, along with the results of Müller, Schlage-Puchta and Roichman, demons…
We investigate the asymptotical behaviour of the transition probabilities of the simple random walk on the 2-comb. In particular we obtain space-time uniform asymptotical estimates which show the lack of symmetry of this walk better than local limit estimates. Our results also point out the impossibility of getting Jones-type non-Gaussian estimates.
We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ) with an infinite two-sided self-avoiding walk (SAW), which can also be described as the result of glueing together two independent uniform infinite quadrangulations of the half-plane (UIHPQs). We prove a lower bound on the displacement of the SAW which, combined with estimates from our previous paper, shows that the self-avoiding walk is diffusive. As a byproduct this implies that the volume growth exponent of the lattice in question is $4$ (as is the case for the standard UIPQ); nevertheless, using our previous work …
Motivated by problems in behavioural finance, we provide two explicit constructions of a randomized stopping time which embeds a given centered distribution $μ$ on integers into a simple symmetric random walk in a uniformly integrable manner. Our first construction has a simple Markovian structure: at each step, we stop if an independent coin with a state-dependent bias returns tails. Our second construction is a discrete analogue of the celebrated Azéma-Yor solution and requires independent coin tosses only when excursions away from maximum breach predefined levels. Further, this construction…
