Cycles In Random Permutations Research Articles

Distribution of the number of cycles in directed and undirected random regular graphs of degree 2.

We present analytical results for the distribution of the number of cycles in directed and undirected random 2-regular graphs (2-RRGs) consisting of N nodes. In directed 2-RRGs each node has one inbound link and one outbound link, while in undirected 2-RRGs each node has two undirected links. Since all the nodes are of degree k=2, the resulting networks consist of cycles. These cycles exhibit a broad spectrum of lengths, where the average length of the shortest cycle in a random network instance scales with lnN, while the length of the longest cycle scales with N. The number of cycles varies between different network instances in the ensemble, where the mean number of cycles 〈S〉 scales with lnN. Here we present exact analytical results for the distribution P_{N}(S=s) of the number of cycles s in ensembles of directed and undirected 2-RRGs, expressed in terms of the Stirling numbers of the first kind. In both cases the distributions converge to a Poisson distribution in the large N limit. The moments and cumulants of P_{N}(S=s) are also calculated. The statistical properties of directed 2-RRGs are equivalent to the combinatorics of cycles in random permutations of N objects. In this context our results recover and extend known results. In contrast, the statistical properties of cycles in undirected 2-RRGs have not been studied before.

Precise asymptotics of longest cycles in random permutations without macroscopic cycles

We consider Ewens random permutations of length n conditioned to have no cycle longer than nβ with 0<β<1 and study the asymptotic behaviour as n→∞. We obtain very precise information on the joint distribution of the lengths of the longest cycles; in particular we prove a functional limit theorem where the cumulative number of long cycles converges to a Poisson process in the suitable scaling. Furthermore, we prove convergence of the total variation distance between joint cycle counts and suitable independent Poisson random variables up to a significantly larger maximal cycle length than previously known. Finally, we remove a superfluous assumption from a central limit theorem for the total number of cycles proved in an earlier paper.

Tail bounds on hitting times of randomized search heuristics using variable drift analysis

AbstractDrift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics (RSHs) such as evolutionary algorithms (EAs), simulated annealing,etc. The vast majority of existing drift theorems yield bounds on the expected value of the hitting time for a target state, for example the set of optimal solutions, without making additional statements on the distribution of this time. We address this lack by providing a general drift theorem that includes bounds on the upper and lower tail of the hitting time distribution. The new tail bounds are applied to prove very precise sharp-concentration results on the running time of a simple EA on standard benchmark problems, including the class of general linear functions. On all these problems, the probability of deviating by anr-factor in lower-order terms of the expected time decreases exponentially withr. The usefulness of the theorem outside the theory of RSHs is demonstrated by deriving tail bounds on the number of cycles in random permutations. All these results handle a position-dependent (variable) drift that was not covered by previous drift theorems with tail bounds. Finally, user-friendly specializations of the general drift theorem are given.

Coverage fluctuations in theater models

We introduce the theater model, which is the simplest variant of directed random sequential adsorption in one dimension with point source and steric interactions. Particles enter sequentially an initially empty row of L sites and adsorb irreversibly at randomly chosen places. If two particles occupy adjacent sites, they prevent further particles from passing them. A jammed configuration without available empty sites is eventually reached. More generally, we investigate the class of models parametrized by b, the number of consecutive particles needed to form a blockage. We show analytically that the occupations of different sites in jammed configurations exhibit long-range correlations obeying scaling laws, for all integers , so that the total number of particles grows as a subextensive power of L, with exponent , and keeps fluctuating even for very large systems. The exactly known relative number variance measuring this lack of self-averaging is maximal for the theater model stricto sensu (b = 2). In the special case where b = 1, so that each adsorbed particle is a blockage, the model can be mapped onto the statistics of records in sequences of random variables and of cycles in random permutations. A two-sided variant of the model is also considered. In both situations the number of particles grows only logarithmically with L, and it is self-averaging.

The number of cycles in random permutations without long cycles is asymptotically Gaussian

The number of cycles in random permutations without long cycles is asymptotically Gaussian

Large cycles in random permutations related to the Heisenberg model

We study the weighted version of the interchange process where a permutation receives weight $\theta^{\#\mathrm{cycles}}$. For $\theta=2$ this is Tóth's representation of the quantum Heisenberg ferromagnet on the complete graph. We prove, for $\theta>1$, that large cycles appear at "low temperature".

Cycle structure of random permutations with cycle weights

We investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the parameters, while the distributions of finite cycles are usually independent Poisson random variables.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 109-133, 2014

Counts of Failure Strings in Certain Bernoulli Sequences

In a sequence of independent Bernoulli trials the probability for success in the kth trial is p k , k = 1, 2, …. The number of strings with a given number of failures between two subsequent successes is studied. Explicit expressions for distributions and moments are obtained for the case in which p k = a/(a + b + k − 1), a &amp;gt; 0, b ≥ 0. Also, the limit behaviour of the longest failure string in the first n trials is considered. For b = 0, the strings correspond to cycles in random permutations.

Counts of Failure Strings in Certain Bernoulli Sequences

In a sequence of independent Bernoulli trials the probability for success in thekth trial ispk,k= 1, 2, …. The number of strings with a given number of failures between two subsequent successes is studied. Explicit expressions for distributions and moments are obtained for the case in whichpk=a/(a+b+k− 1),a&gt; 0,b≥ 0. Also, the limit behaviour of the longest failure string in the firstntrials is considered. Forb= 0, the strings correspond to cycles in random permutations.

Limit Measures Arising in the Asympyotic Theory of Symmetric Groups. I.

Limit Measures Arising in the Asympyotic Theory of Symmetric Groups. I.