我们构造了一个有界度图$G$，使得其上的简单随机游动是瞬时的，但是随机游动路径（即，随机游动所穿过的所有边的子图）几乎肯定地仅具有1/4个割点。我们还证明了任何瞬态马尔可夫链的截点的期望个数是无穷的。这回答了James，Lyons和Peres的两个问题[A Transient马尔可夫链With Actually Many Cutpoints（2007）Festschrift for David Freedman]。此外，我们考虑有限连通图$G$上的简单随机游动，它从某个固定顶点$x$开始，当它第一次访问另一个固定顶点$y$时停止。我们提供了一个下界之间的预期有效阻力$x$和$y$在步行的路径，给出了部分回答[安。35（2007）732- 738]。
英文摘要：
We construct a bounded degree graph $G$, such that a simple random walk on it is transient but the random walk path (i.e., the subgraph of all the edges the random walk has crossed) has only finitely many cutpoints, almost surely. We also prove that the expected number of cutpoints of any transient Markov chain is infinite. This answers two questions of James, Lyons and Peres [A Transient Markov Chain With Finitely Many Cutpoints (2007) Festschrift for David Freedman]. Additionally, we consider a simple random walk on a finite connected graph $G$ that starts at some fixed vertex $x$ and is stopped when it first visits some other fixed vertex $y$. We provide a lower bound on the expected effective resistance between $x$ and $y$ in the path of the walk, giving a partial answer to a question raised in [Ann. Probab. 35 (2007) 732--738].
We prove that a simple random walk on quasi-transitive graphs with the volume growth being faster than any polynomial of degree 4 has a.s. infinitely many cut times, and hence infinitely many cutpoints. This confirms a conjecture raised by I. Benjamini, O. Gurel-Gurevich and O. Schramm [2011, Cutpoints and resistance of random walk paths, {\it Ann. Probab.} {\bf 39(3)}, 1122-1136] that PATH of simple random walk on any transient vertex-transitive graph has a.s. infinitely many cutpoints in the corresponding case.
We give conditions under which near-critical stochastic processes on the half-line have infinitely many or finitely many cutpoints, generalizing existing results on nearest-neighbour random walks to adapted processes with bounded increments satisfying appropriate conditional increment moments conditions. We apply one of these results to deduce that a class of transient zero-drift Markov chains in $\mathbb{R}^d$, $d \geq 2$, possess infinitely many separating annuli, generalizing previous results on spatially homogeneous random walks.
In this paper, we study (1,2) and (2,1) random walks in varying environments on the lattice of positive half line. We assume that the transition probabilities at site $n$ are asymptotically constants as $n\rightarrow\infty.$ For (1,2) random walk, we get some elaborate asymptotic behaviours of various escape probabilities and hitting probabilities of the walk. Such observations and some delicate analysis of continued fractions and the product of nonnegative matrices enable us to give criteria for finiteness of the number of cutpoints of both (1,2) and (2,1) random walks, which generalize E. Cs…
We prove Barlow--Bass type resistance estimates for two random walks associated with repeated barycentric subdivisions of a triangle. If the random walk jumps between the centers of triangles in the subdivision that have common sides, the resistance scales as a power of a constant $ρ$ which is theoretically estimated to be in the interval $5/4\leqslantρ\leqslant3/2$, with a numerical estimate $ρ\approx1.306$. This corresponds to the theoretical estimate of spectral dimension $d_S$ between 1.63 and 1.77, with a numerical estimate $d_S\approx1.74$. On the other hand, if the random walk jumps bet…
In this article, universal concentration estimates are established for the local times of random walks on weighted graphs in terms of the resistance metric. As a particular application of these, a modulus of continuity for local times is provided in the case when the graphs in question satisfy a certain volume growth condition with respect to the resistance metric. Moreover, it is explained how these results can be applied to self-similar fractals, for which they are shown to be useful for deriving scaling limits for local times and asymptotic bounds for the cover time distribution.
A new proof is given for the formula for the expected return time of a random walk on a graph. This proof makes use of known relationships between electric resistance and random walks.
We produce a characterization of finite metric spaces which are given by the effective resistance of a graph. This characterization is applied to the more general context of resistance metrics defined by Kigami. A countably infinite resistance metric gives rise to a sequence of finite, increasing graphs with invariant effective resistance. We show that these graphs have a unique limit graph in terms of the convergence of edge weights and that their associated random walks converge weakly to the random walk on the limit graph. If the limit graph is recurrent, its effective resistance is identif…
We prove expectation and concentration results for the following random variables on an Erdős-Rényi random graph $\mathcal{G}\left(n,p\right)$ in the sparsely connected regime $\log n + \log\log \log n \leq np < n^{1/10}$: effective resistances, random walk hitting and commute times, the Kirchoff index, cover cost, random target times, the mean hitting time and Kemeny's constant. For the effective resistance between two vertices our concentration result extends further to $np\geq c\log n, \; c>0$. To achieve these results, we show that a strong connectedness property holds with high probabilit…
We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial location of the random walk in question. The limiting processes are the analogues of those for higher-dimensional versions of the model, but additional logarithmic terms in the scaling factors are needed to see these. The proof applies recently developed machinery relating the scaling of resistance metric spaces and stochastic processes, with key inputs being n…
We completely characterize when the free effective resistance of an infinite graph can be expressed in terms of simple hitting probabilities of the graphs random walk.
We study the simple random walk on the Uniform Infinite Half-Plane Map, which is the local limit of critical Boltzmann planar maps with a large and simple boundary. We prove that the simple random walk is recurrent, and that the resistance between the root and the boundary of the hull of radius $r$ is at least of order $\log r$. This resistance bound is expected to be sharp, and is better than those following from previous proofs of recurrence for non bounded-degree planar maps models. Our main tools are the self-duality of uniform planar maps, a classical lemma about duality of resistances an…
We study the trace of the incipient infinite oriented branching random walk in $\mathbb{Z}^d \times \mathbb{Z}_+$ when the dimension is $d = 6$. Under suitable moment assumptions, we show that the electrical resistance between the root and level $n$ is $O(n \log^{-ξ}n )$ for a $ξ> 0$ that does not depend on details of the model.
We establish that if a sequence of spaces equipped with resistance metrics and measures converge with respect to the Gromov-Hausdorff-vague topology, and a certain non-explosion condition is satisfied, then the associated stochastic processes also converge. This result generalises previous work on trees, fractals, and various models of random graphs. We further conjecture that it will be applicable to the random walk on the incipient infinite cluster of critical bond percolation on the high-dimensional integer lattice.
We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d. increments or a two-dimensional Brownian motion via a "mating-of-trees" type bijection. This class includes the uniform infinite planar triangulation (UIPT), the infinite-volume limits of random planar maps weighted by the number of spanning trees, bipolar orientations, or Schnyder woods they admit, and the $γ$-mated-CRT map for $γ\in (0,2)$. For each of these maps, we obtain an upper bound for the Green's function on the diagonal, an upper bound for the effective re…
Given any $γ>0$ and for $η=\{η_v\}_{v\in \mathbb Z^2}$ denoting a sample of the two-dimensional discrete Gaussian free field on $\mathbb Z^2$ pinned at the origin, we consider the random walk on~$\mathbb Z^2$ among random conductances where the conductance of edge $(u, v)$ is given by $\mathrm{e}^{γ(η_u + η_v)}$. We show that, for almost every~$η$, this random walk is recurrent and that, with probability tending to~1 as $T\to \infty$, the return probability at time~$2T$ decays as $T^{-1+o(1)}$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from…
We introduce partial loop-erasing operators. We show that by applying a refinement sequence of partial loop-erasing operators to a finite Markov chain, we get a process equivalent to the chronological loop-erased Markov chain. As an application, we construct loop-erased random paths on bounded domains of resistance spaces as the weak limit of the loop erasure of the Markov chains on a sequence of finite sets approximating the space, and the limit is independent of the approximating sequences. The random paths we constructed are simple paths almost surely, and they can be viewed as the loop-era…
In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree $n$ with an independent copy of a graph $G_n$ and gluing the inserted graphs along the tree structure. We assume that there exist constants $d, R \geq 1, v < \infty$ such that the diameter, effective resistance across and volume of $G_n$ respectively grow like $n^{\frac{1}{d}}, n^{\frac{1}{R}}, n^v$ as $n \to \infty$. We also assume that the underlying Galton-Watson tree is critical with offspring tails decaying like $cx^{-α}$ for some constant…
Looptrees have recently arisen in the study of critical percolation on the uniform infinite planar triangulation. Here we consider random infinite looptrees defined as the local limit of the looptree associated with a critical Galton--Watson tree conditioned to be large. We study simple random walk on these infinite looptrees by means of providing estimates on volume and resistance growth. We prove that if the offspring distribution of the Galton--Watson process is in the domain of attraction of a stable distribution with index $α\in(1,2]$ then the spectral dimension of the looptree is $2α/(α+…
Recently, Kigami's resistance form framework has been applied to provide a general approach for deriving the scaling limits of random walks on graphs with a fractal scaling limit. As an illustrative example, this article describes an application to the random conductance model with heavy tails on nested fractal graphs.
In certain instances an electric network transforms in natural ways by the addition or removal of an edge. This can have interesting consequences for random walks, in light of the known relationships between electric resistance and random walks. We exhibit several instances in which this can be used to prove facts or simplify calculations. In particular, a new proof is given for the formula for the expected return time of a random walk on a graph. We also show how hitting times can be calculated in certain instances when a network differs from a highly symmetric one by one edge.
The scaling limit of random walk and the intrinsic metric on planar critical percolation
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge（剑桥大学纯数学与数理统计系）；Faculty of Mathematics, University of Vienna（维也纳大学数学学院）
In a previous paper [arXiv:1604.05440], we studied certain random walks on the hyperbolic graphs $X$ associated with the self-similar sets $K$, and showed that the discrete energy ${\mathcal E}_X$ on $X$ has an induced energy form ${\mathcal E}_K$ on $K$ that is a Gagliardo-type integral. The domain of ${\mathcal E}_K$ is a Besov space $Λ^{α, β/2}_{2,2}$ where $α$ is the Hausdorff dimension of $K$ and $β$ is a parameter determined by the "return ratio" of the random walk. In this paper, we study the functional relationship of ${\mathcal E}_X$ and ${\mathcal E}_K$. In particular, we investigate…
We investigate the validity of the "Einstein relations" in the general setting of unimodular random networks. These are equalities relating scaling exponents: $d_w = d_f + \tildeζ$ and $d_s = 2 d_f/d_w$, where $d_w$ is the walk dimension, $d_f$ is the fractal dimension, $d_s$ is the spectral dimension, and $\tildeζ$ is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that if $d_f$ and $\tildeζ \geq 0$ exist, then $d_w$ and $d_s$ exist, and the aforementioned …
We give an $m^{1+o(1)}β^{o(1)}$-time algorithm for generating a uniformly random spanning tree in an undirected, weighted graph with max-to-min weight ratio $β$. We also give an $m^{1+o(1)}ε^{-o(1)}$-time algorithm for generating a random spanning tree with total variation distance $ε$ from the true uniform distribution. Our second algorithm's runtime does not depend on the edge weights. Our $m^{1+o(1)}β^{o(1)}$-time algorithm is the first almost-linear time algorithm for the problem --- even on unweighted graphs --- and is the first subquadratic time algorithm for sparse weighted graphs. Our …
