The Einstein relation for random walks on graphs

Some relations between the spectra of simple and non-backtracking random walks

We establish some relations between the spectra of simple and non-backtracking random walks on non-regular graphs, generalizing some well-known facts for regular graphs. Our two main results are 1) a quantitative relation between the mixing rates of the simple random walk and of the non-backtracking random walk 2) a variant of the "Ihara determinant formula" which expresses the characteristic polynomial of the adjacency matrix, or of the laplacian, as the determinant of a certain non-backtracking random walk with holomorphic weights.

动态电导上的恒速随机游走

The constant speed random walk on dynamical conductances

本文研究动态随机电导中的恒速有偏随机游走，证明其在特定参数范围内为弹道式，并在全参数范围内建立退火与淬火中心极限定理，同时指出不可逆性导致经典爱因斯坦关系失效。

Limit theorems for random walks in dynamic random environment

We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties and concentration inequalities for the environment as seen from the position of the walker, i.e the environment process. We also obtain ergodicity of the uniquely ergodic measure of the environment process as well as continuity as a function of the jump rates of the walker. As a consequence we obtain several limit theorems, such as law of large numbers, Einstein relation, central limit theorem and concentration propert…

度量测度空间上的爱因斯坦关系

The Einstein Relation on Metric Measure Spaces

本文在度量测度空间上回顾了联系Hausdorff、局部游走和谱维数的爱因斯坦关系，证明了三种分形维数在双Lipschitz同构下的不变性，并探讨了非Lipschitz映射对维数的扭曲影响。

Balls-in-boxes duality for coalescing random walks and coalescing Brownian motions

We present a duality relation between two systems of coalescing random walks and an analogous duality relation between two systems of coalescing Brownian motions. Our results extends previous work in the literature and we apply it to the study of a system of coalescing Brownian motions with Poisson immigration.

An invariance principle for random walk bridges conditioned to stay positive

We prove an invariance principle for the bridge of a random walk conditioned to stay positive, when the random walk is in the domain of attraction of a stable law, both in the discrete and in the absolutely continuous setting. This includes as a special case the convergence under diffusive rescaling of random walk excursions toward the normalized Brownian excursion, for zero mean, finite variance random walks. The proof exploits a suitable absolute continuity relation together with some local asymptotic estimates for random walks conditioned to stay positive, recently obtained by Vatutin and W…

动态渗流上的有偏随机游走

Biased random walk on dynamical percolation

本文研究$\mathbb{Z}^d$上动态渗流的有偏随机游走，利用再生时间建立大数定律和不变原理，验证Einstein关系，并发现$d \geq 2$时速度非单调，由临界曲线分隔出严格递增或递减的两个参数区域。

The relation between quenched and annealed Lyapunov exponents in random potential on trees

In the first part of the article our subject of interest is a simple symmetric random walk on the integers which faces a random risk to be killed. This risk is described by random potentials, which in turn are defined by a sequence of independent and identically distributed non-negative random variables. To determine the risk of taking a walk in these potentials we consider the decay of the Green function. There are two possible tools to describe this decay: The quenched Lyapunov exponents and the annealed Lyapunov exponents. It turns out that on the integers we can state a precise relation be…

Asymptotic behaviour of the simple random walk on the 2-comb

We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in $n$ steps, proving that for all these quantities the order is $n^{1/4}$ for the horizontal projection and $n^{1/2}$ for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by $n^{1/4}$ and $n^{1/2}$ the horizontal and vertical…

Supermultiplicative relations in models of interacting self-avoiding walks and polygons

Fekete's lemma shows the existence of limits in subadditive sequences. This lemma, and generalisations of it, also have been used to prove the existence of thermodynamic limits in statistical mechanics. In this paper it is shown that the two variable supermultiplicative relation $$ p_{n_1}(m_1)\,p_{n_2}(m_2) \leq p_{n_1+n_2}(m_1 + m_2) $$ together with mild assumptions, imply the existence of the limit $$ \log \mathcal{P}_\#(ε) = \lim_{n\to\infty} \frac{1}{n} \log p_n(\lfloor εn \rfloor).$$ This is a generalisation of Fekete's lemma. The existence of $\mathcal{P}_\#(ε)$ are proven for models o…

Random walks on random horospheric products

By developing the entropy theory of random walks on equivalence relations and analyzing the asymptotic geometry of horospheric products we describe the Poisson boundary for random walks on random horospheric products of trees.

On the relation between quantum walks and zeta functions

We present an explicit formula for the characteristic polynomial of the transition matrix of the discrete-time quantum walk on a graph via the second weighted zeta function. As applications, we obtain new proofs for the results on spectra of the transition matrix and its positive support.

从Kreweras游走至渗流三角剖分的分支边界过程

From Kreweras walks to branching perimeter processes of percolated triangulations

本文通过Bernardi等人的结果建立三角剖分渗流界面剥离过程边界与Kreweras游走的几何关系，扩展至分支剥离探索，揭示了生长-碎裂过程与相关布朗游程的联系。

关于量子游走与绝对 zeta 函数之间的关系

On the relation between quantum walks and absolute zeta functions

本文建立图上 Grover walk 与绝对 zeta 函数的联系，证明量子游走诱导的 zeta 函数构成依赖边数的绝对自守形式，并以循环图为例将其绝对 zeta 函数表示为二阶多重 gamma 函数。