2,527 research outputs found
Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs
We study two of the simple rules on finite graphs under the death-birth
updating and the imitation updating discovered by Ohtsuki, Hauert, Lieberman
and Nowak [Nature 441 (2006) 502-505]. Each rule specifies a payoff-ratio
cutoff point for the magnitude of fixation probabilities of the underlying
evolutionary game between cooperators and defectors. We view the Markov chains
associated with the two updating mechanisms as voter model perturbations. Then
we present a first-order approximation for fixation probabilities of general
voter model perturbations on finite graphs subject to small perturbation in
terms of the voter model fixation probabilities. In the context of regular
graphs, we obtain algebraically explicit first-order approximations for the
fixation probabilities of cooperators distributed as certain uniform
distributions. These approximations lead to a rigorous proof that both of the
rules of Ohtsuki et al. are valid and are sharp.Comment: Published in at http://dx.doi.org/10.1214/12-AAP849 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Limit laws for discrete excursions and meanders and linear functional equations with a catalytic variable
We study limit distributions for random variables defined in terms of
coefficients of a power series which is determined by a certain linear
functional equation. Our technique combines the method of moments with the
kernel method of algebraic combinatorics. As limiting distributions the area
distributions of the Brownian excursion and meander occur. As combinatorial
applications we compute the area laws for discrete excursions and meanders with
an arbitrary finite set of steps and the area distribution of column convex
polyominoes. As a by-product of our approach we find the joint distribution of
area and final altitude for meanders with an arbitrary step set, and for
unconstrained Bernoulli walks (and hence for Brownian Motion) the joint
distribution of signed areas and final altitude. We give these distributions in
terms of their moments.Comment: 33 pages, 1 figur
Semigroups, rings, and Markov chains
We analyze random walks on a class of semigroups called ``left-regular
bands''. These walks include the hyperplane chamber walks of Bidigare, Hanlon,
and Rockmore. Using methods of ring theory, we show that the transition
matrices are diagonalizable and we calculate the eigenvalues and
multiplicities. The methods lead to explicit formulas for the projections onto
the eigenspaces. As examples of these semigroup walks, we construct a random
walk on the maximal chains of any distributive lattice, as well as two random
walks associated with any matroid. The examples include a q-analogue of the
Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are
``generalized derangement numbers'', which may be of independent interest.Comment: To appear in J. Theoret. Proba
Computing periods of rational integrals
A period of a rational integral is the result of integrating, with respect to
one or several variables, a rational function over a closed path. This work
focuses particularly on periods depending on a parameter: in this case the
period under consideration satisfies a linear differential equation, the
Picard-Fuchs equation. I give a reduction algorithm that extends the
Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs
equations. The resulting algorithm is elementary and has been successfully
applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at
http://pierre.lairez.fr/supp/periods
Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration
The field of analytic combinatorics, which studies the asymptotic behaviour
of sequences through analytic properties of their generating functions, has led
to the development of deep and powerful tools with applications across
mathematics and the natural sciences. In addition to the now classical
univariate theory, recent work in the study of analytic combinatorics in
several variables (ACSV) has shown how to derive asymptotics for the
coefficients of certain D-finite functions represented by diagonals of
multivariate rational functions. We give a pedagogical introduction to the
methods of ACSV from a computer algebra viewpoint, developing rigorous
algorithms and giving the first complexity results in this area under
conditions which are broadly satisfied. Furthermore, we give several new
applications of ACSV to the enumeration of lattice walks restricted to certain
regions. In addition to proving several open conjectures on the asymptotics of
such walks, a detailed study of lattice walk models with weighted steps is
undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page
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