An invariance principle for biased voter model interfaces

亚弹道区域随机电导中偏置随机游走的淬火不变原理

Quenched invariance principle for biased random walks in random conductances in the sub-ballistic regime

研究$d\geq 5$维正随机电导中偏置随机游走，在亚弹道区域证明其重标度后淬火收敛于分数动力学。

Genealogies in Expanding Populations

The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [25]. The first step, which is easily accomplished using results of Mueller and Tribe [38], is to show that when space and time are rescaled correctly, our biased voter model converges to a Wright-Fisher SPDE. A simple extension of a result of Durrett and Restrepo [18] then shows that the dual branching coalescing random walk converges to a branching Brownian motion in which particles coalesce after an exponentially distribu…

Biased Consensus Dynamics on Regular Expander Graphs

Consensus protocols play an important role in the study of distributed algorithms. In this paper, we study the effect of bias on two popular consensus protocols, namely, the {\em voter rule} and the {\em 2-choices rule} with binary opinions. We assume that agents with opinion $1$ update their opinion with a probability $q_1$ strictly less than the probability $q_0$ with which update occurs for agents with opinion $0$. We call opinion $1$ as the superior opinion and our interest is to study the conditions under which the network reaches consensus on this opinion. We assume that the agents are l…

Invariance principle for biased Boostrap Random Walk

Our main goal is to study a class of processes whose increments are generated via a cellular automata rule. Given the increments of a simple biased random walk, a new sequence of (dependent) Bernoulli random variables is produced. It is built, from the original sequence, according to a cellular automata rule. Equipped with these two sequences, we construct two more according to the same cellular automata rule. The construction is repeated a fixed number of times yielding an infinite array ($\{-K,\ldots,K\}\times\N$) of (dependent) Bernoulli random variables. %In turn, using this new sequence a…

Limit of 1-Dimensional Mixed-Mechanism Interacting Particle System Model

Elaborating on the model from voter process with mixed-mechanism under suitable scaling, I have two new mechanisms which are random switch and unbiased local Homogenization and subtly biased advantage but with state dependent coefficient involved. The most crucial one, the existence of high-frequency duplication generating the diffusion term and noise term in each case identifies the limit equation as SPDE driven by space time white noise.

Central limit theorems for biased randomly trapped random walks on Z

We prove CLTs for biased randomly trapped random walks in one dimension. In particular, we will establish an annealed invariance principal by considering a sequence of regeneration times under the assumption that the trapping times have finite second moment. In a quenched environment, an environment dependent centring is determined which is necessary to achieve a central limit theorem. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive and prove a tight bound on the bias required to obtain such limiting behaviour.

An integral characterization of the Dirichlet process

We give a new integral characterization of the Dirichlet process on a general phase space. To do so we first prove a characterization of the nonsymmetric Beta distribution via size-biased sampling. Two applications are a new characterization of the Dirichlet distribution and a marked version of a classical characterization of the Poisson-Dirichlet distribution via invariance under size-biased sampling.

大规模随机成对交互网络系统中基于不变性原理的集中结果

An invariance principle based concentration result for large-scale stochastic pairwise interaction network systems

该研究针对含个体突变与邻居成对交互的随机网络系统，借助李雅普诺夫型函数推导其平稳分布的集中结果，突破完全混合模型限制，适用于Erdős–Rényi随机图等更广网络结构。

Extreme value statistics of positive recurrent centrally biased random walks

We consider the extreme value statistics of centrally-biased random walks with asymptotically-zero drift in the ergodic regime. We fully characterize the asymptotic distribution of the maximum for this class of Markov chains lacking translational invariance, with a particular emphasis on the relation between the time scaling of the expected value of the maximum and the stationary distribution of the process.

两个反向偏置 Wiener 过程的合并

Merge of two oppositely biased Wiener processes

提出将两个反向偏置 Brownian 运动合并为单一正则过程的技术，结果遵循含常扩散系数和非线性漂移的随机微分方程，新过程具有均方位移时空平移不变性且可通过位置依赖随机游走高效模拟。

树上随机环境中随机游走的不变性原理

Invariance principles for random walks in random environment on trees

本文利用电阻度量测度空间的泛函极限定理，研究树上随机环境随机游走的标度极限，证明有偏游走与边增强游走分别收敛为 CRT 上随机 Gaussian 势中的 Brown 运动。