arXiv subjects

Oliver Redner

Publications and source records attributed to Oliver Redner.

5 recordsLinked to original sources

arXiv2004

Discrete Approximation of Non-Compact Operators Describing Continuum-of-Alleles Models

We consider the eigenvalue equation for the largest eigenvalue of certain kinds of non-compact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite rank, which constitutes a discretization procedure. For this purpose, two standard methods of approximation theory, the Nyström and the Galerkin method, are generalized. The operators considered describe models for mutation and selection of an infinitely large population of individuals that are labeled by real numbers, commonly called continuum-of-alleles (COA) models.

math.SP↗

arXiv2004

Unequal Crossover Dynamics in Discrete and Continuous Time

We analyze a class of models for unequal crossover (UC) of sequences containing sections with repeated units that may differ in length. In these, the probability of an `imperfect' alignment, in which the shorter sequence has d units without a partner in the longer one, scales like q^d as compared to `perfect' alignments where all these copies are paired. The class is parameterized by this penalty factor q. An effectively infinite population size and thus deterministic dynamics is assumed. For the extreme cases q=0 and q=1, and any initial distribution whose moments satisfy certain conditions, we prove the convergence to one of the known fixed points, uniquely determined by the mean copy number, in both discrete and continuous time. For the intermediate parameter values, the existence of fixed points is shown.

math.DS↗

arXiv2002

Mutation-Selection Balance: Ancestry, Load, and Maximum Principle

We show how concepts from statistical physics, such as order parameter, thermodynamic limit, and quantum phase transition, translate into biological concepts in mutation-selection models for sequence evolution and can be used there. The article takes a biological point of view within a population genetics framework, but contains an appendix for physicists, which makes this correspondence clear. We analyze the equilibrium behavior of deterministic haploid mutation-selection models. Both the forward and the time-reversed evolution processes are considered. The stationary state of the latter is called the ancestral distribution, which turns out as a key for the study of mutation-selection balance. We find that it determines the sensitivity of the equilibrium mean fitness to changes in the fitness values and discuss implications for the evolution of mutational robustness. We further show that the difference between the ancestral and the population mean fitness, termed mutational loss, provides a measure for the sensitivity of the equilibrium mean fitness to changes in the mutation rate. For a class of models in which the number of mutations in an individual is taken as the trait value, and fitness is a function of the trait, we use the ancestor formulation to derive a simple maximum principle, from which the mean and variance of fitness and the trait may be derived; the results are exact for a number of limiting cases, and otherwise yield approximations which are accurate for a wide range of parameters. These results are applied to (error) threshold phenomena caused by the interplay of selection and mutation. They lead to a clarification of concepts, as well as criteria for the existence of thresholds.

cond-mat.stat-mech↗

arXiv2000

Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

We demonstrate that the invaded cluster algorithm, recently introduced by Machta et al, is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic Ising models in two dimensions. The algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits. On two quasiperiodic graphs which were not investigated in this respect before, the twelvefold symmetric square-triangle tiling and the tenfold symmetric Tübingen triangle tiling, we determine the critical temperature. Furthermore, a generalization of the algorithm to non-identical coupling strengths is presented and applied to a class of Ising models on the Labyrinth tiling. For generic cases in which the heuristic Harris-Luck criterion predicts deviations from the Onsager universality class, we find a magnetic critical exponent different from the Onsager value. But also notable exceptions to the criterion are found which consist not only of the exactly solvable cases, in agreement with a recent exact result, but also of the self-dual ones and maybe more.

cond-mat.stat-mech↗

arXiv1998

Graphical representations and cluster algorithms for critical points with fields

A two-replica graphical representation and associated cluster algorithm is described that is applicable to ferromagnetic Ising systems with arbitrary fields. Critical points are associated with the percolation threshold of the graphical representation. Results from numerical simulations of the Ising model in a staggered field are presented. The dynamic exponent for the algorithm is measured to be less than 0.5.

cond-mat.stat-mech↗