Evidence for a continuum limit in causal set dynamics

We find evidence for a continuum limit of a particular causal set dynamics which depends on only a single ``coupling constant'' $p$ and is easy to simulate on a computer. The model in question is a stochastic process that can also be interpreted as 1-dimensional directed percolation, or in terms of random graphs.Comment: 24 pages, 19 figures, LaTeX, adjusted terminolog

Non-central generalized q-factorial coefficients and q-Stirling numbers

AbstractThe qs-differences of the non-central generalized q-factorials of t of order n, scale parameter s and non-centrality parameter r, at t=0, are thoroughly examined. These numbers for s→0 and s→∞ converge to the non-central q-Stirling numbers of the first and the second kind, respectively. Explicit expressions, recurrence relations, generating functions and other properties of these q-numbers are derived. Further, a sequence of Bernoulli trials is considered in which the conditional probability of success at the nth trial, given that k successes occur before that trial, varies geometrically with n and k. Then, the probability functions of the number of successes in n trials and the number of trials until the occurrence of the kth success are deduced in terms of the qs-differences of the non-central generalized q-factorials of t of order n, scale parameter s and non-centrality parameter r

Markov Processes Involving q-Stirling Numbers

In this paper we consider the Markov process defined by P 1;1 = 1; P n;` = (1 \Gamma n;` ) \Delta P n\Gamma1;` + n;`\Gamma1 \Delta P n\Gamma1;`\Gamma1 for transition probabilities n;` = q ` and n;` = q n\Gamma1 . We give closed forms for the distributions and the moments of the underlying random variables. Thereby we observe that the distributions can be easily described in terms of q--Stirling numbers of the second kind. Their occurrence in a purely time dependent Markov process allows a natural approximation for these numbers through the normal distribution. We also show that these Markov processes describe some parameters related to the study of random graphs as well as to the analysis of algorithms. Keywords: q--Stirling numbers, Markov processes, random graphs, approximate counting. Contents 1 Introduction 4 2 Some Definitions 5 3 The Markov Process 7 4 Approximations 14 5 Concluding Remarks 17 1 Introduction In this paper we will study the Markov process defined by P..