scholarly journals A non-local random walk on the hypercube

2017 ◽  
Vol 49 (4) ◽  
pp. 1288-1299
Author(s):  
Evita Nestoridi
Keyword(s):  
Random Walk ◽  
Mixing Time ◽  
Non Local

Abstract In this paper we study the random walk on the hypercube (ℤ / 2ℤ)n which at each step flips k randomly chosen coordinates. We prove that the mixing time for this walk is of the order (n / k)logn. We also prove that if k = o(n) then the walk exhibits cutoff at (n / 2k)logn with window n / 2k.

scholarly journals Uniform mixing time for random walk on lamplighter graphs

2014 ◽  
Vol 50 (4) ◽  
pp. 1140-1160 ◽  
Author(s):  
Júlia Komjáthy ◽  
Jason Miller ◽  
Yuval Peres
Keyword(s):  
Random Walk ◽  
Mixing Time

scholarly journals Assessing significance in a Markov chain without mixing

2017 ◽  
Vol 114 (11) ◽  
pp. 2860-2864 ◽  
Author(s):  
Maria Chikina ◽  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Stationary Distribution ◽  
Potential Application ◽  
Null Hypothesis ◽  
Value Function ◽  
Mixing Time ◽  
Statistical Test ◽  
Reversible Markov Chain ◽  
A Value

We present a statistical test to detect that a presented state of a reversible Markov chain was not chosen from a stationary distribution. In particular, given a value function for the states of the Markov chain, we would like to show rigorously that the presented state is an outlier with respect to the values, by establishing a p value under the null hypothesis that it was chosen from a stationary distribution of the chain. A simple heuristic used in practice is to sample ranks of states from long random trajectories on the Markov chain and compare these with the rank of the presented state; if the presented state is a 0.1% outlier compared with the sampled ranks (its rank is in the bottom 0.1% of sampled ranks), then this observation should correspond to a p value of 0.001. This significance is not rigorous, however, without good bounds on the mixing time of the Markov chain. Our test is the following: Given the presented state in the Markov chain, take a random walk from the presented state for any number of steps. We prove that observing that the presented state is an ε-outlier on the walk is significant at p=2ε under the null hypothesis that the state was chosen from a stationary distribution. We assume nothing about the Markov chain beyond reversibility and show that significance at p≈ε is best possible in general. We illustrate the use of our test with a potential application to the rigorous detection of gerrymandering in Congressional districting.

scholarly journals Random walks on generating sets for finite groups

10.37236/1322 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
F. R. K. Chung ◽  
R. L. Graham
Keyword(s):  
Finite Group ◽  
Random Walk ◽  
Random Walks ◽  
Finite Groups ◽  
Mixing Time ◽  
Cartesian Product ◽  
Generating Sets ◽  
Random Elements ◽  
A Finite Group

We analyze a certain random walk on the cartesian product $G^n$ of a finite group $G$ which is often used for generating random elements from $G$. In particular, we show that the mixing time of the walk is at most $c_r n^2 \log n$ where the constant $c_r$ depends only on the order $r$ of $G$.

scholarly journals The Mixing Time for a Random Walk on the Symmetric Group Generated by Random Involutions

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Megan Bernstein
Keyword(s):  
Information Theory ◽  
Random Walk ◽  
Symmetric Group ◽  
Random Walks ◽  
Binomial Distribution ◽  
Unitary Group ◽  
Mixing Time ◽  
Sufficient Time ◽  
International Audience ◽  
A New Technique

International audience The involution walk is a random walk on the symmetric group generated by involutions with a number of 2-cycles sampled from the binomial distribution with parameter p. This is a parallelization of the lazy transposition walk onthesymmetricgroup.Theinvolutionwalkisshowninthispapertomixfor1 ≤p≤1fixed,nsufficientlylarge 2 in between log1/p(n) steps and log2/(1+p)(n) steps. The paper introduces a new technique for finding eigenvalues of random walks on the symmetric group generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. This is especially efficient at calculating the large eigenvalues. The smaller eigenvalues are handled by developing monotonicity relations that also give after sufficient time the likelihood order, the order from most likely to least likely state. The walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of back holes.

scholarly journals A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis

2016 ◽  
Author(s):  
Andreas Buttenschön ◽  
Thomas Hillen ◽  
Alf Gerisch ◽  
Kevin J. Painter
Keyword(s):  
Random Walk ◽  
Mean Field ◽  
Polarization Vector ◽  
Jump Process ◽  
Biological Properties ◽  
Cellular Adhesion ◽  
Cell Polarization ◽  
Local Models ◽  
A Cell ◽  
Non Local

AbstractCellular adhesion provides one of the fundamental forms of biological interaction between cells and their surroundings, yet the continuum modelling of cellular adhesion has remained mathematically challenging. In 2006, Armstrong et al. proposed a mathematical model in the form of an integro-partial differential equation. Although successful in applications, a derivation from an underlying stochastic random walk has remained elusive. In this work we develop a framework by which non-local models can be derived from a space-jump process. We show how the notions of motility and a cell polarization vector can be naturally included. With this derivation we are able to include microscopic biological properties into the model. We show that particular choices yield the original Armstrong model, while others lead to more general models, including a doubly non-local adhesion model and non-local chemotaxis models. Finally, we use random walk simulations to confirm that the corresponding continuum model represents the mean field behaviour of the stochastic random walk.

scholarly journals Mean-field models for segregation dynamics

2020 ◽  
pp. 1-22
Author(s):  
MARTIN BURGER ◽  
JAN-FREDERIK PIETSCHMANN ◽  
HELENE RANETBAUER ◽  
CHRISTIAN SCHMEISER ◽  
MARIE-THERESE WOLFRAM
Keyword(s):  
Random Walk ◽  
Mean Field ◽  
Single Species ◽  
Random Motion ◽  
Local Effects ◽  
Mean Field Models ◽  
Existence And Stability ◽  
Non Local ◽  
The Rich ◽  
Multiple Species

In this paper, we derive and analyse mean-field models for the dynamics of groups of individuals undergoing a random walk. The random motion of individuals is only influenced by the perceived densities of the different groups present as well as the available space. All individuals have the tendency to stay within their own group and avoid the others. These interactions lead to the formation of aggregates in case of a single species and to segregation in the case of multiple species. We derive two different mean-field models, which are based on these interactions and weigh local and non-local effects differently. We discuss existence and stability properties of solutions for both models and illustrate the rich dynamics with numerical simulations.

Sign in / Sign up

Export Citation Format

Share Document