68 research outputs found
Large-deviation properties of the extended Moran model
The distributions of the times to the first common ancestor t_mrca is
numerically studied for an ecological population model, the extended Moran
model. This model has a fixed population size N. The number of descendants is
drawn from a beta distribution Beta(alpha, 2-alpha) for various choices of
alpha. This includes also the classical Moran model (alpha->0) as well as the
uniform distribution (alpha=1). Using a statistical mechanics-based
large-deviation approach, the distributions can be studied over an extended
range of the support, down to probabilities like 10^{-70}, which allowed us to
study the change of the tails of the distribution when varying the value of
alpha in [0,2]. We find exponential distributions p(t_mrca)~ delta^{t_mrca} in
all cases, with systematically varying values for the base delta. Only for the
cases alpha=0 and alpha=1, analytical results are known, i.e.,
delta=\exp(-2/N^2) and delta=2/3, respectively. We recover these values,
confirming the validity of our approach. Finally, we also study the
correlations between t_mrca and the number of descendants.Comment: 8 pages, 8 figure
Statistics of Branched Populations Split into Different Types
Some population is made of n individuals that can be of P possible species (or types) at equilibrium. How are individuals scattered among types? We study two random scenarios of such species abundance distributions. In the first one, each species grows from independent founders according to a Galton-Watson branching process. When the number of founders P is either fixed or random (either Poisson or geometrically-distributed), a question raised is: given a population of n individuals as a whole, how does it split into the species types? This model is one pertaining to forests of Galton-Watson trees. A second scenario that we will address in a similar way deals with forests of increasing trees. Underlying this setup, the creation/annihilation of clusters (trees) is shown to result from a recursive nucleation/aggregation process as one additional individual is added to the total population
On Discrete-Time Multiallelic Evolutionary Dynamics Driven by Selection
We revisit some problems arising in the context of multiallelic discrete-time evolutionary dynamics driven by fitness. We consider both the deterministic and the stochastic setups and for the latter both the Wright-Fisher and the Moran approaches. In the deterministic formulation, we construct a Markov process whose Master equation identifies with the nonlinear
deterministic evolutionary equation. Then, we draw the attention on a class of fitness matrices that plays some role in the important matter of polymorphism: the class of strictly ultrametric fitness matrices. In the random cases, we focus on fixation probabilities, on various conditionings on nonfixation, and on (quasi)stationary distributions
A Bose-Einstein Approach to the Random Partitioning of an Integer
Consider N equally-spaced points on a circle of circumference N. Choose at
random n points out of on this circle and append clockwise an arc of
integral length k to each such point. The resulting random set is made of a
random number of connected components. Questions such as the evaluation of the
probability of random covering and parking configurations, number and length of
the gaps are addressed. They are the discrete versions of similar problems
raised in the continuum. For each value of k, asymptotic results are presented
when n,N both go to infinity according to two different regimes. This model may
equivalently be viewed as a random partitioning problem of N items into n
recipients. A grand-canonical balls in boxes approach is also supplied, giving
some insight into the multiplicities of the box filling amounts or spacings.
The latter model is a k-nearest neighbor random graph with N vertices and kn
edges. We shall also briefly consider the covering problem in the context of a
random graph model with N vertices and n (out-degree 1) edges whose endpoints
are no more bound to be neighbors
A necklace of Wulff shapes
In a probabilistic model of a film over a disordered substrate, Monte-Carlo
simulations show that the film hangs from peaks of the substrate. The film
profile is well approximated by a necklace of Wulff shapes. Such a necklace can
be obtained as the infimum of a collection of Wulff shapes resting on the
substrate. When the random substrate is given by iid heights with exponential
distribution, we prove estimates on the probability density of the resulting
peaks, at small density
Information and (co-)variances in discrete evolutionary genetics involving solely selection
The purpose of this Note is twofold: First, we introduce the general
formalism of evolutionary genetics dynamics involving fitnesses, under both the
deterministic and stochastic setups, and chiefly in discrete-time. In the
process, we particularize it to a one-parameter model where only a selection
parameter is unknown. Then and in a parallel manner, we discuss the estimation
problems of the selection parameter based on a single-generation frequency
distribution shift under both deterministic and stochastic evolutionary
dynamics. In the stochastics, we consider both the celebrated Wright-Fisher and
Moran models.Comment: a paraitre dans Journal of Statistical Mechanics: Theory and
Application
Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation
A detailed study is presented for a large class of uncoupled continuous-time
random walks (CTRWs). The master equation is solved for the Mittag-Leffler
survival probability. The properly scaled diffusive limit of the master
equation is taken and its relation with the fractional diffusion equation is
discussed. Finally, some common objections found in the literature are
thoroughly reviewed.Comment: Preprint version of an already published paper. 8 page
Nonextensivity and multifractality in low-dimensional dissipative systems
Power-law sensitivity to initial conditions at the edge of chaos provides a
natural relation between the scaling properties of the dynamics attractor and
its degree of nonextensivity as prescribed in the generalized statistics
recently introduced by one of us (C.T.) and characterized by the entropic index
. We show that general scaling arguments imply that , where and are the
extremes of the multifractal singularity spectrum of the attractor.
This relation is numerically checked to hold in standard one-dimensional
dissipative maps. The above result sheds light on a long-standing puzzle
concerning the relation between the entropic index and the underlying
microscopic dynamics.Comment: 12 pages, TeX, 4 ps figure
A Pearson-Dirichlet random walk
A constrained diffusive random walk of n steps and a random flight in Rd,
which can be expressed in the same terms, were investigated independently in
recent papers. The n steps of the walk are identically and independently
distributed random vectors of exponential length and uniform orientation.
Conditioned on the sum of their lengths being equal to a given value l,
closed-form expressions for the distribution of the endpoint of the walk were
obtained altogether for any n for d=1, 2, 4 . Uniform distributions of the
endpoint inside a ball of radius l were evidenced for a walk of three steps in
2D and of two steps in 4D. The previous walk is generalized by considering step
lengths which are distributed over the unit (n-1) simplex according to a
Dirichlet distribution whose parameters are all equal to q, a given positive
value. The walk and the flight above correspond to q=1. For any d >= 3, there
exist, for integer and half-integer values of q, two families of
Pearson-Dirichlet walks which share a common property. For any n, the d
components of the endpoint are jointly distributed as are the d components of a
vector uniformly distributed over the surface of a hypersphere of radius l in a
space Rk whose dimension k is an affine function of n for a given d. Five
additional walks, with a uniform distribution of the endpoint in the inside of
a ball, are found from known finite integrals of products of powers and Bessel
functions of the first kind. They include four different walks in R3 and two
walks in R4. Pearson-Liouville random walks, obtained by distributing the total
lengths of the previous Pearson-Dirichlet walks, are finally discussed.Comment: 33 pages 1 figure, the paper includes the content of a recently
submitted work together with additional results and an extended section on
Pearson-Liouville random walk
Exact Scale Invariance in Mixing of Binary Candidates in Voting Model
We introduce a voting model and discuss the scale invariance in the mixing of
candidates. The Candidates are classified into two categories
and are called as `binary' candidates. There are in total
candidates, and voters vote for them one by one. The probability that a
candidate gets a vote is proportional to the number of votes. The initial
number of votes (`seed') of a candidate is set to be . After
infinite counts of voting, the probability function of the share of votes of
the candidate obeys gamma distributions with the shape exponent
in the thermodynamic limit . Between the
cumulative functions of binary candidates, the power-law relation
with the critical exponent
holds in the region . In the double
scaling limit and with
fixed, the relation holds
exactly over the entire range . We study the data on
horse races obtained from the Japan Racing Association for the period 1986 to
2006 and confirm scale invariance.Comment: 19 pages, 8 figures, 2 table
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