arXiv subjects
Publications and source records attributed to Perla Sousi.
We study the scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in '86 for the volume of the range in dimension two.
Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This fails for directed chains, as the biased random walk on the cycle $\mathbb{Z}_n$ shows. However, we establish that for Eulerian digraphs, the mixing time is $O(mn)$, where $m$ is the number of edges and $n$ is the number of vertices. In the reversible case, the mixing time is robust to the change of the laziness parameter. Surprisingly, in the directed setting the mixing time can be sensitive to such changes. We also study exploration and cover times for random walks on Eulerian digraphs and prove universal upper bounds in analogy to the undirected case.
We study the capacity of the range of a transient simple random walk on $\mathbb{Z}^d$. Our main result is a central limit theorem for the capacity of the range for $d\ge 6$. We present a few open questions in lower dimensions.
Let $P$ be the transition matrix of a finite, irreducible and reversible Markov chain. We say the continuous time Markov chain $X$ has transition matrix $P$ and speed $λ$ if it jumps at rate $λ$ according to the matrix $P$. Fix $λ_X,λ_Y,λ_Z\geq 0$, then let $X,Y$ and $Z$ be independent Markov chains with transition matrix $P$ and speeds $λ_X,λ_Y$ and $λ_Z$ respectively, all started from the stationary distribution. What is the chance that $X$ and $Y$ meet before either of them collides with $Z$? For each choice of $λ_X,λ_Y$ and $λ_Z$ with $\max(λ_X,λ_Y)>0$, we prove a lower bound for this probability which is uniform over all transitive, irreducible and reversible chains. In the case that $λ_X=λ_Y=1$ and $λ_Z=0$ we prove a strengthening of our main theorem using a martingale argument. We provide an example showing the transitivity assumption cannot be removed for general $λ_X,λ_Y$ and $λ_Z$.
Suppose X and Y are two independent irreducible Markov chains on n states. We consider the intersection time, which is the first time their trajectories intersect. We show for reversible and lazy chains that the total variation mixing time is always upper bounded by the expected intersection time taken over the worst starting states. For random walks on trees we show the two quantities are equivalent. We obtain an expression for the expected intersection time in terms of the eigenvalues for reversible and transitive chains. For such chains we also show that it is up to constants the geometric mean of n and E[I], where I is the number of intersections up to the uniform mixing time. Finally for random walks on regular graphs we obtain sharp inequalities that relate the expected intersection time to maximum hitting time and mixing time.
Suppose that $(X,Y,Z)$ is a random walk in $\mathbb{Z}^3$ that moves in the following way: on the first visit to a vertex only $Z$ changes by $\pm 1$ equally likely, while on later visits to the same vertex $(X,Y)$ performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.
Let $X$ be a fractional Brownian motion in $\mathbb{R}^d$. For any Borel function $f:[0,1] \to \mathbb{R}^d$, we express the Hausdorff dimension of the image and the graph of $X+f$ in terms of $f$. This is new even for the case of Brownian motion and continuous $f$, where it was known that this dimension is almost surely constant. The expression involves an adaptation of the parabolic dimension previously used by Taylor and Watson to characterize polarity for the heat equation. In the case when the graph of $f$ is a self-affine McMullen-Bedford carpet, we obtain an explicit formula for the dimension of the graph of $X+f$ in terms of the generating pattern. In particular, we show that it can be strictly bigger than the maximum of the Hausdorff dimension of the graph of $f$ and that of $X$. Despite the random perturbation, the Minkowski and Hausdorff dimension of the graph of $X+f$ can disagree.
Suppose that $X$ is a simple random walk on $\Z_n^d$ for $d \geq 3$ and, for each $t$, we let $\U(t)$ consist of those $x \in \Z_n^d$ which have not been visited by $X$ by time $t$. Let $\tcov$ be the expected amount of time that it takes for $X$ to visit every site of $\Z_n^d$. We show that there exists $0 < α_0(d) \leq α_1(d) < 1$ and a time $t_* = \tcov(1+o(1))$ as $n \to \infty$ such that the following is true. For $α> α_1(d)$ (resp.\ $α< α_0(d)$), the total variation distance between the law of $\U(αt_*)$ and the law of i.i.d.\ Bernoulli random variables indexed by $\Z_n^d$ with success probability~$n^{-αd}$ tends to~$0$ (resp.\ $1$) as $n \to \infty$. Let $τ_α$ be the first time $t$ that $|\U(t)| = n^{d-αd}$. We also show that the total variation distance between the law of $\U(τ_α)$ and the law of a uniformly chosen set from $\Z_n^d$ with size $n^{d-αd}$ tends to $0$ (resp.\ $1$) for $α> α_1(d)$ (resp.\ $α< α_0(d)$) as $n \to \infty$.
We construct a family of trees on which a lazy simple random walk exhibits total variation cutoff. The main idea behind the construction is that hitting times of large sets should be concentrated around their means. For this sequence of trees we compute the mixing time, the relaxation time and the cutoff window.
We consider irreducible reversible discrete time Markov chains on a finite state space. Mixing times and hitting times are fundamental parameters of the chain. We relate them by showing that the mixing time of the lazy chain is equivalent to the maximum over initial states x and large sets A of the hitting time of A starting from x. We also prove that the first time when averaging over two consecutive time steps is close to stationarity is equivalent to the mixing time of the lazy version of the chain.
Let $σ$ be a permutation of $\{0,\ldots,n\}$. We consider the Markov chain $X$ which jumps from $k\neq 0,n$ to $σ(k+1)$ or $σ(k-1)$, equally likely. When $X$ is at 0 it jumps to either $σ(0)$ or $σ(1)$ equally likely, and when $X$ is at $n$ it jumps to either $σ(n)$ or $σ(n-1)$, equally likely. We show that the identity permutation maximizes the expected hitting time of n, when the walk starts at 0. More generally, we prove that the hitting time of a random walk on a strongly connected $d$-directed graph is maximized when the graph is the line $[0,n]\cap\Z$ with $d-2$ self-loops at every vertex and $d-1$ self-loops at 0 and $n$.
We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger (1976) and can be interpreted as involving symmetric rearrangements of domains around infinity. As applications, we prove two comparison results for general Levy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Levy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Levy motions are symmetrically rearranged. This essentially generalizes an isoperimetric inequality of Peres and Sousi (2011) for the Wiener sausage. In the second application, we show that the q-capacity of a Borel measurable set for a Levy process can only decrease if the set and the Levy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe (1983) for symmetric Levy processes.
We consider irreducible Markov chains on a finite state space. We show that the mixing time of any such chain is equivalent to the maximum, over initial states $x$ and moving large sets $(A_s)_s$, of the hitting time of $(A_s)_s$ starting from $x$. We prove that in the case of the $d$-dimensional torus the maximum hitting time of moving targets is equal to the maximum hitting time of stationary targets. Nevertheless, we construct a transitive graph where these two quantities are not equal, resolving an open question of Aldous and Fill on a "cat and mouse" game.
We study fractal properties of the image and the graph of Brownian motion in $\R^d$ with an arbitrary c{à}dl{à}g drift $f$. We prove that the Minkowski (box) dimension of both the image and the graph of $B+f$ over $A\subseteq [0,1]$ are a.s.\ constants. We then show that for all $d\geq 1$ the Minkowski dimension of $(B+f)(A)$ is at least the maximum of the Minkowski dimension of $f(A)$ and that of $B(A)$. We also prove analogous results for the graph. For linear Brownian motion, if the drift $f$ is continuous and $A=[0,1]$, then the corresponding inequality for the graph is actually an equality.
A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle $\Z_n$. A hunter and a rabbit move on the nodes of $\Z_n$ without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al (2003) provide strategies for hunter and rabbit that are optimal up to constant factors and achieve probability of capture in the first $n$ steps of order $1/\log n$. We show these strategies yield a Kakeya set consisting of $4n$ triangles with minimal area, (up to constant), namely $Θ(1/\log n)$. As far as we know, this is the first non-iterative construction of a boundary-optimal Kakeya set. Considering the continuum analog of the game yields a construction of a random Kakeya set from two independent standard Brownian motions $\{B(s): s \ge 0\}$ and $\{W(s): s \ge 0\}$. Let $τ_t:=\min\{s \ge 0: B(s)=t\}$. Then $X_t=W(τ_t)$ is a Cauchy process, and $K:=\{(a,X_t+at) : a,t \in [0,1]\}$ is a Kakeya set of zero area. The area of the $ε$-neighborhood of $K$ is as small as possible, i.e., almost surely of order $Θ(1/|\log ε|)$.
Let $μ_1,... μ_k$ be $d$-dimensional probability measures in $\R^d$ with mean 0. At each step we choose one of the measures based on the history of the process and take a step according to that measure. We give conditions for transience of such processes and also construct examples of recurrent processes of this type. In particular, in dimension 3 we give the complete picture: every walk generated by two measures is transient and there exists a recurrent walk generated by three measures.
Let the nodes of a Poisson point process move independently in $\R^d$ according to Brownian motions. We study the isolation time for a target particle that is placed at the origin, namely how long it takes until there is no node of the Poisson point process within distance $r$ of it. In the case when the target particle does not move, we obtain asymptotics for the tail {probability} which are tight up to constants in the exponent in dimension $d\geq 3$ and tight up to logarithmic factors in the exponent for dimensions $d=1,2$. In the case when the target particle is allowed to move independently of the Poisson point process, we show that the best strategy for the target to avoid isolation is to stay put.
Let $(ξ(s))_{s\geq 0}$ be a standard Brownian motion in $d\geq 1$ dimensions and let $(D_s)_{s \geq 0}$ be a collection of open sets in $\R^d$. For each $s$, let $B_s$ be a ball centered at 0 with $\vol(B_s) = \vol(D_s)$. We show that $\E[\vol(\cup_{s \leq t}(ξ(s) + D_s))] \geq \E[\vol(\cup_{s \leq t}(ξ(s) + B_s))]$, for all $t$. In particular, this implies that the expected volume of the Wiener sausage increases when a drift is added to the Brownian motion.