arXiv subjects

Richard Arratia

Publications and source records attributed to Richard Arratia.

25 records · Page 2Linked to original sources

arXiv2013

On the Amount of Dependence in the Prime Factorization of a Uniform Random Integer

How much dependence is there in the prime factorization of a random integer distributed uniformly from 1 to n? How much dependence is there in the decomposition into cycles of a random permutation of n points? What is the relation between the Poisson-Dirichlet process and the scale invariant Poisson process? These three questions have essentially the same answers, with respect to total variation distance, considering only small components, and with respect to a Wasserstein distance, considering all components. The Wasserstein distance is the expected number of changes -- insertions and deletions -- needed to change the dependent system into an independent system. In particular we show that for primes, roughly speaking, 2+o(1) changes are necessary and sufficient to convert a uniformly distributed random integer from 1 to n into a random integer prod_{p leq n} p^{Z_p} in which the multiplicity Z_p of the factor p is geometrically distributed, with all Z_p independent. The changes are, with probability tending to 1, one deletion, together with a random number of insertions, having expectation 1+o(1). The crucial tool for showing that 2+epsilon suffices is a coupling of the infinite independent model of prime multiplicities, with the scale invariant Poisson process on (0,infty). A corollary of this construction is the first metric bound on the distance to the Poisson-Dirichlet in Billingsley's 1972 weak convergence result. Our bound takes the form: there are couplings in which E sum |log P_i(n) - (log n) V_i | = O(\log \log n), where P_i denotes the i-th largest prime factor and V_i denotes the i-th component of the Poisson-Dirichlet process. It is reasonable to conjecture that O(1) is achievable.

math.NT↗

arXiv2012

On the singularity of random Bernoulli matrices - novel integer partitions and lower bound expansions

We prove a lower bound expansion on the probability that a random $\pm 1$ matrix is singular, and conjecture that such expansions govern the actual probability of singularity. These expansions are based on naming the most likely, second most likely, and so on, ways that a Bernoulli matrix can be singular; the most likely way is to have a null vector of the form $e_i \pm e_j$, which corresponds to the integer partition 11, with two parts of size 1. The second most likely way is to have a null vector of the form $e_i \pm e_j \pm e_k \pm e_\ell$, which corresponds to the partition 1111. The fifth most likely way corresponds to the partition 21111. We define and characterize the "novel partitions" which show up in this series. As a family, novel partitions suffice to detect singularity, i.e., any singular Bernoulli matrix has a left null vector whose underlying integer partition is novel. And, with respect to this property, the family of novel partitions is minimal. We prove that the only novel partitions with six or fewer parts are 11, 1111, 21111, 111111, 221111, 311111, and 322111. We prove that there are fourteen novel partitions having seven parts. We formulate a conjecture about which partitions are "first place and runners up," in relation to the Erdős-Littlewood-Offord bound. We prove some bounds on the interaction between left and right null vectors.

math.PR↗

arXiv2010

Size bias, sampling, the waiting time paradox, and infinite divisibility: when is the increment independent?

With $X^*$ denoting a random variable with the $X$-size bias distribution, what are all distributions for $X$ such that it is possible to have $X^*=X+Y$, $Y\geq 0$, with $X$ and $Y$ {\em independent}? We give the answer, due to Steutel \cite{steutel}, and also discuss the relations of size biasing to the waiting time paradox, renewal theory, sampling, tightness and uniform integrability, compound Poisson distributions, infinite divisibility, and the lognormal distributions.

math.PR↗

arXiv2005

Local central limit theorems, the high-order correlations of rejective sampling and logistic likelihood asymptotics

Let I_1,...,I_n be independent but not necessarily identically distributed Bernoulli random variables, and let X_n=\sum_{j=1}^nI_j. For νin a bounded region, a local central limit theorem expansion of P(X_n=EX_n+ν) is developed to any given degree. By conditioning, this expansion provides information on the high-order correlation structure of dependent, weighted sampling schemes of a population E (a special case of which is simple random sampling), where a set d\subset E is sampled with probability proportional to \prod_{A\in d}x_A, where x_A are positive weights associated with individuals A\in E. These results are used to determine the asymptotic information, and demonstrate the consistency and asymptotic normality of the conditional and unconditional logistic likelihood estimator for unmatched case-control study designs in which sets of controls of the same size are sampled with equal probability.

math.ST↗

arXiv2005

How likely is an i.i.d. degree sequence to be graphical?

Given i.i.d. positive integer valued random variables D_1,...,D_n, one can ask whether there is a simple graph on n vertices so that the degrees of the vertices are D_1,...,D_n. We give sufficient conditions on the distribution of D_i for the probability that this be the case to be asymptotically 0, {1/2} or strictly between 0 and {1/2}. These conditions roughly correspond to whether the limit of nP(D_i\geq n) is infinite, zero or strictly positive and finite. This paper is motivated by the problem of modeling large communications networks by random graphs.

math.PR↗

arXiv2004

The Interlace Polynomial of a Graph

Motivated by circle graphs, and the enumeration of Euler circuits, we define a one-variable ``interlace polynomial'' for any graph. The polynomial satisfies a beautiful and unexpected reduction relation, quite different from the cut and fuse reduction characterizing the Tutte polynomial. It emerges that the interlace graph polynomial may be viewed as a special case of the Martin polynomial of an isotropic system, which underlies its connections with the circuit partition polynomial and the Kauffman brackets of a link diagram. The graph polynomial, in addition to being perhaps more broadly accessible than the Martin polynomial for isotropic systems, also has a two-variable generalization that is unknown for the Martin polynomial. We consider extremal properties of the interlace polynomial, its values for various special graphs, and evaluations which relate to basic graph properties such as the component and independence numbers.

math.CO↗

arXiv2004

A Two-Variable Interlace Polynomial

We introduce a new graph polynomial in two variables. This ``interlace'' polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the significant differences are that our expansion is over vertex rather than edge subsets, and the rank and nullity employed are those of an adjacency matrix rather than an incidence matrix. The second computation is by a three-term reduction formula involving a graph pivot; the pivot arose previously in the study of interlacement and Euler circuits in four-regular graphs. We consider a few properties and specializations of the two-variable interlace polynomial. One specialization, the ``vertex-nullity interlace polynomial'', is the single-variable interlace graph polynomial we studied previously, closely related to the Tutte-Martin polynomial on isotropic systems previously considered by Bouchet. Another, the ``vertex-rank interlace polynomial'', is equally interesting. Yet another specialization of the two-variable polynomial is the independent-set polynomial.

math.CO↗