Sum rules for effective resistances in infinite graphs

After the Explosion: Dirichlet Forms and Boundary Problems for Infinite Graphs

Formal Laplace operators are analyzed for a large class of resistance networks with vertex weights. The graphs are completed with respect to the minimal resistance path metric. Compactness and a novel connectivity hypothesis for the completed graphs play an essential role. A version of the Dirichlet problem is solved. Self adjoint Laplace operators and the probability semigroups they generate are constructed using reflecting and absorbing conditions on subsets of the graph boundary.

Percolation and disorder-resistance in cellular automata

We rigorously prove a form of disorder-resistance for a class of one-dimensional cellular automaton rules, including some that arise as boundary dynamics of two-dimensional solidification rules. Specifically, when started from a random initial seed on an interval of length $L$, with probability tending to one as $L\to\infty$, the evolution is a replicator. That is, a region of space-time of density one is filled with a spatially and temporally periodic pattern, punctuated by a finite set of other finite patterns repeated at a fractal set of locations. On the other hand, the same rules exhibit …

Electrical resistance of the low dimensional critical branching random walk

We show that the electrical resistance between the origin and generation n of the incipient infinite oriented branching random walk in dimensions d<6 is O(n^{1-alpha}) for some universal constant alpha>0. This answers a question of Barlow, Járai, Kumagai and Slade [2].

Resistance boundaries of infinite networks

A resistance network is a connected graph $(G,c)$. The conductance function $c_{xy}$ weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form $\mathcal E$ produces a Hilbert space structure ${\mathcal H}_{\mathcal E}$ on the space of functions of finite energy. The relationship between the natural Dirichlet form $\mathcal E$ and the discrete Laplace operator $Δ$ on a finite network is given by $\mathcal E(u,v) = \la u, \Lap v\ra_2$, where the latter is the usual $\ell^2$ inner product. We describe a reproducing kernel $\{v_x\}$ for $\…

一维临界长程渗流有效电阻的多项式下界

Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation

本文研究一维临界长程渗流模型的电网络性质，证明对所有 β>0，从原点或区间到远处区域的有效电阻均具有多项式下界，从而排除了 β=1 附近存在相变的可能性。

Positive speed for high-degree automaton groups

Mother groups are the basic building blocks for polynomial automaton groups. We show that, in contrast with mother groups of degree 0 or 1, any bounded, symmetric, generating random walk on the mother groups of degree at least 3 has positive speed. The proof is based on an analysis of resistance in fractal mother graphs. We give upper bounds on resistances in these graphs, and show that infinite versions are tran- sient.

Cutpoints and resistance of random walk paths

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 sto…

Coupling any number of balls in the infinite-bin model

The infinite-bin model, introduced by Foss and Konstantopoulos, describes the Markovian evolution of configurations of balls placed inside bins, obeying certain transition rules. We prove that we can couple the behaviour of any finite number of balls, provided at least two different transition rules are allowed. This coupling makes it possible to define the regeneration events needed by Foss and Zachary to prove convergence results for the distribution of the balls.

Local ergodicity in the exclusion process on an infinite weighted graph

We establish an abstract local ergodic theorem, under suitable space-time scaling, for the (boundary-driven) symmetric exclusion process on an increasing sequence of balls covering an infinite weighted graph. The proofs are based on 1-block and 2-blocks estimates utilizing the resistance structure of the graph; the moving particle lemma established recently by the author; and discrete harmonic analysis. Our ergodic theorem applies to any infinite weighted graph upon which random walk is strongly recurrent in the sense of Barlow, Delmotte, and Telcs; these include many trees, fractal graphs, an…