scholarly journals Non Unitary Random Walks

2009 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Philippe Jacquet
Keyword(s):  
Black Hole ◽  
Random Walk ◽  
Random Walks ◽  
Transition Probabilities ◽  
Surprising Result ◽  
Energy Effect ◽  
Information Loss Paradox ◽  
International Audience ◽  
Probability Weight ◽  
Interesting Analogy

International audience Motivated by the recent refutation of information loss paradox in black hole by Hawking, we investigate the new concept of {\it non unitary random walks}. In a non unitary random walk, we consider that the state 0, called the {\it black hole}, has a probability weight that decays exponentially in $e^{-\lambda t}$ for some $\lambda>0$. This decaying probabilities affect the probability weight of the other states, so that the the apparent transition probabilities are affected by a repulsion factor that depends on the factors $\lambda$ and black hole lifetime $t$. If $\lambda$ is large enough, then the resulting transition probabilities correspond to a neutral random walk. We generalize to {\it non unitary gravitational walks} where the transition probabilities are function of the distance to the black hole. We show the surprising result that the black hole remains attractive below a certain distance and becomes repulsive with an exactly reversed random walk beyond this distance. This effect has interesting analogy with so-called dark energy effect in astrophysics.

Random walk in a random environment with correlated sites

2001 ◽  
Vol 38 (4) ◽  
pp. 1018-1032 ◽  
Author(s):  
T. Komorowski ◽  
G. Krupa
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Field ◽  
Random Environment ◽  
Law Of Large Numbers ◽  
Transition Probabilities ◽  
Direct Application ◽  
Random Environments ◽  
Large Numbers ◽  
Stationary Random Field

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Zd. The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

scholarly journals A further study of an approximation for last-exit and first-passage probabilities of a random walk

1994 ◽  
Vol 7 (3) ◽  
pp. 411-422
Author(s):  
D. J. Daley ◽  
L. D. Servi
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Transition Probabilities ◽  
Continuous Lattice ◽  
First Passage ◽  
Continuity Properties ◽  
Exit Probabilities

Identities between first-passage or last-exit probabilities and unrestricted transition probabilities that hold for left- or right-continuous lattice-valued random walks form the basis of an intuitively based approximation that is demonstrated by computation to hold for certain random walks without either the left- or right-continuity properties. The argument centers on the use of ladder variables; the identities are known to hold asymptotically from work of Iglehart leading to Brownian meanders.

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 Stochastic LU factorizations, Darboux transformations and urn models

2018 ◽  
Vol 55 (3) ◽  
pp. 862-886 ◽  
Author(s):  
F. Alberto Grünbaum ◽  
Manuel D. de la Iglesia
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Free Parameter ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Urn Models ◽  
Triangular Matrices ◽  
New Family ◽  
Step Transition

Abstract We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

scholarly journals Non-crossing trees revisited: cutting down and spanning subtrees

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Alois Panholzer
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Limiting Distributions ◽  
Brownian Excursion ◽  
International Audience ◽  
Two Parameters

International audience Here we consider two parameters for random non-crossing trees: $\textit{(i)}$ the number of random cuts to destroy a size-$n$ non-crossing tree and $\textit{(ii)}$ the spanning subtree-size of $p$ randomly chosen nodes in a size-$n$ non-crossing tree. For both quantities, we are able to characterise for $n → ∞$ the limiting distributions. Non-crossing trees are almost conditioned Galton-Watson trees, and it has been already shown, that the contour and other usually associated discrete excursions converge, suitable normalised, to the Brownian excursion. We can interpret parameter $\textit{(ii)}$ as a functional of a conditioned random walk, and although we do not have such an interpretation for parameter $\textit{(i)}$, we obtain here limiting distributions, that are also arising as limits of some functionals of conditioned random walks.

scholarly journals Random Walks in Hypergraph

2021 ◽  
Vol 15 ◽  
pp. 13-20
Author(s):  
Abdelghani Bellaachia ◽  
Mohammed Al-Dhelaan
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Transition Probabilities ◽  
Graph Representation ◽  
Real World Data ◽  
Data Set ◽  
Random Walks On Graphs ◽  
Text Ranking ◽  
Relationship Structure ◽  
The Relationship

Random walks on graphs have been extensively used for a variety of graph-based problems such as ranking vertices, predicting links, recommendations, and clustering. However, many complex problems mandate a high-order graph representation to accurately capture the relationship structure inherent in them. Hypergraphs are particularly useful for such models due to the density of information stored in their structure. In this paper, we propose a novel extension to defining random walks on hypergraphs. Our proposed approach combines the weights of destination vertices and hyperedges in a probabilistic manner to accurately capture transition probabilities. We study and analyze our generalized form of random walks suitable for the structure of hypergraphs. We show the effectiveness of our model by conducting a text ranking experiment on a real world data set with a 9% to 33% improvement in precision and a range of 7% to 50% improvement in Bpref over other random walk approaches.

scholarly journals Rate of Escape of Random Walks on Regular Languages and Free Products by Amalgamation of Finite Groups

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Lorenz A. Gilch
Keyword(s):  
Random Walks ◽  
Finite Groups ◽  
Transition Probabilities ◽  
Finite Alphabet ◽  
Free Products ◽  
Rate Of Escape ◽  
International Audience ◽  
Special Case ◽  
Current Word ◽  
Natural Way

International audience We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.

scholarly journals Frogs and some other interacting random walks models

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Serguei Yu. Popov
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Active Particle ◽  
Open Problems ◽  
Simple Version ◽  
Random Walks On Graphs ◽  
Interacting Random Walks ◽  
International Audience ◽  
Frog Model

International audience We review some recent results for a system of simple random walks on graphs, known as \emphfrog model. Also, we discuss several modifications of this model, and present a few open problems. A simple version of the frog model can be described as follows: There are active and sleeping particles living on some graph. Each active particle performs a simple random walk with discrete time and at each moment it may disappear with probability 1-p. When an active particle hits a sleeping particle, the latter becomes active.

scholarly journals Discrete Random Walks on One-Sided ``Periodic'' Graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michael Drmota
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Generating Functions ◽  
Connected Graph ◽  
Infinite Graphs ◽  
Reflected Brownian Motion ◽  
Strongly Connected ◽  
International Audience ◽  
Null Recurrence

International audience In this paper we consider discrete random walks on infinite graphs that are generated by copying and shifting one finite (strongly connected) graph into one direction and connecting successive copies always in the same way. With help of generating functions it is shown that there are only three types for the asymptotic behaviour of the random walk. It either converges to the stationary distribution or it can be approximated in terms of a reflected Brownian motion or by a Brownian motion. In terms of Markov chains these cases correspond to positive recurrence, to null recurrence, and to non recurrence.

scholarly journals Deterministic Random Walks on the Integers

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Joshua Cooper ◽  
Benjamin Doerr ◽  
Joel Spencer ◽  
Gábor Tardos
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Deterministic Process ◽  
One Dimensional ◽  
Space And Time ◽  
Dimensional Version ◽  
International Audience ◽  
The One

International audience We analyze the one-dimensional version of Jim Propp's $P$-machine, a simple deterministic process that simulates a random walk on $\mathbb{Z}$. The "output'' of the machine is astonishingly close to the expected behavior of a random walk, even on long intervals of space and time.

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