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The remarkable universality of the eigenvalue correlation functions is perhaps one of the most salient findings in random matrix theory. Particularly for short-range separations of the eigenvalues, the correlation functions have been shown…
In population genetics, extant samples are usually used for inference of past population genetic forces. With the Kingman coalescent and the backward diffusion equation, inference of the marginal likelihood proceeds from an extant sample…
We consider and reformulate a recent definition of multiplication between distributions. We show that this definition can be adopted, in particular, to prove biorthonormality of some distributions arising when looking to the (generalized)…
Natural selection explains how life has evolved over millions of years from more primitive forms. The speed at which this happens, however, has sometimes defied formal explanations when based on random (uniformly distributed) mutations.…
The random variable $1+z_1+z_1z_2+\dots$ appears in many contexts and was shown by Kesten to exhibit a heavy tail distribution. We consider natural extensions of this variable and its associated recursion to $N \times N$ matrices either…
Textual analysis of typical microbial genomes reveals that they have the statistical characteristics of a DNA sequence of a much shorter length. This peculiar property supports an evolutionary model in which a genome evolves by random…
The presence of clusters of rare codons is known to negatively impact the efficiency and accuracy of protein production. In this paper, we demonstrate a statistical method of identifying such clusters in the coding sequence of a gene. Using…
A possible application of the evolution equation for the truncated Mellin moments to determination of the parton distributions in the nucleon is presented. We find that the reconstruction of the initial parton densities at scale $Q_0^2$…
When an infectious disease outbreak is of a relatively small size, describing the ancestry of a sample of infected individuals is difficult because most ancestral models assume large population sizes. Given a set of infected individuals, we…
We show that the empirical distribution of the eigenvalues of the sample covariance matrix of certain random vectors (not necessarily independent entries) with bounded marginal $L^{4}$ norms converges weakly to a compound free Poisson…
Longitudinal molecular data of rapidly evolving viruses and pathogens provide information about disease spread and complement traditional surveillance approaches based on case count data. The coalescent is used to model the genealogy that…
We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function.
The distributed genome hypothesis states that the set of genes in a population of bacteria is distributed over all individuals that belong to the specific taxon. It implies that certain genes can be gained and lost from generation to…
The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths),…
The time evolution of a bounded quantum system is considered in the framework of the orthogonal, unitary and symplectic circular ensembles of random matrix theory. For an $N$ dimensional Hilbert space we prove that in the large $N$ limit…
Using numerical exact diagonalization, we study matrix elements of a local spin operator in the eigenbasis of two different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented…
In this paper, a new mixed Poisson distribution is introduced. This new distribution is obtained by utilizing mixing process, with Poisson distribution as mixed distribution and Transmuted Exponential distribution as mixing distribution.…
The ancestral sequence reconstruction problem is the inference, back in time, of the properties of common sequence ancestors from measured properties of contemporary populations. Standard algorithms for this problem assume independent…
The Poisson distribution arises naturally when dealing with data involving counts, and it has found many applications in inverse problems and imaging. In this work, we develop an approximate Bayesian inference technique based on expectation…
Locations of DNA replication initiation in prokaryotes, called "origins of replication", are well-characterized. However, a mechanistic understanding of the sequence-dependence of the local unzipping of double-stranded DNA, the first step…