scholarly journals Isoperimetric inequalities and mixing time for a random walk on a random point process

2007 ◽  
Vol 17 (5-6) ◽  
pp. 1707-1744 ◽  
Author(s):  
Pietro Caputo ◽  
Alessandra Faggionato
Keyword(s):  
Random Walk ◽  
Point Process ◽  
Mixing Time ◽  
Isoperimetric Inequalities ◽  
Random Point

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 Random point processes and tilings arising from Gaussian analytic functions

2014 ◽  
Vol 70 (a1) ◽  
pp. C523-C523
Author(s):  
Michael Baake ◽  
Holger Koesters ◽  
Robert Moody
Keyword(s):  
Point Process ◽  
Analytic Functions ◽  
Point Processes ◽  
Logarithmic Potential ◽  
Basins Of Attraction ◽  
Random Point ◽  
Specific Class ◽  
Determinantal Point Processes ◽  
Point Sets ◽  
Gaussian Analytic Functions

Getting a grasp of what aperiodic order really entails is going to require collecting and understanding many diverse examples. Aperiodic crystals are at the top of the largely unknown iceberg beneath. Here we present a recently studied form of random point process in the (complex) plane which arises as the sets of zeros of a specific class of analytic functions given by power series with randomly chosen coefficients: Gaussian analytic functions (GAF). These point sets differ from Poisson processes by having a sort of built in repulsion between points, though the resulting sets almost surely fail both conditions of the Delone property. Remarkably the point sets that arise as the zeros of GAFs determine a random point process which is, in distribution, invariant under rotation and translation. In addition, there is a logarithmic potential function for which the zeros are the attractors, and the resulting basins of attraction produce tilings of the plane by tiles which are, almost surely, all of the same area. We discuss GAFs along with their tilings and diffraction, and as well note briefly their relationship to determinantal point processes, which are also of physical interest.

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 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 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 $G$.

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