574 research outputs found
Phase Transition in NK-Kauffman Networks and its Correction for Boolean Irreducibility
In a series of articles published in 1986 Derrida, and his colleagues studied
two mean field treatments (the quenched and the annealed) for
\textit{NK}-Kauffman Networks. Their main results lead to a phase transition
curve () for the
critical average connectivity in terms of the bias of
extracting a "" for the output of the automata. Values of bigger than
correspond to the so-called chaotic phase; while , to an
ordered phase. In~[F. Zertuche, {\it On the robustness of NK-Kauffman networks
against changes in their connections and Boolean functions}. J.~Math.~Phys.
{\bf 50} (2009) 043513], a new classification for the Boolean functions, called
{\it Boolean irreducibility} permitted the study of new phenomena of
\textit{NK}-Kauffman Networks. In the present work we study, once again the
mean field treatment for \textit{NK}-Kauffman Networks, correcting it for {\it
Boolean irreducibility}. A shifted phase transition curve is found. In
particular, for the predicted value by Derrida {\it
et al.} changes to We support our results with
numerical simulations.Comment: 23 pages, 7 Figures on request. Published in Physica D: Nonlinear
Phenomena: Vol.275 (2014) 35-4
Chromatic Zeros On Hierarchical Lattices and Equidistribution on Parameter Space
Associated to any finite simple graph is the chromatic polynomial
whose complex zeroes are called the chromatic zeros of .
A hierarchical lattice is a sequence of finite simple graphs
built recursively using a substitution rule
expressed in terms of a generating graph. For each , let denote the
probability measure that assigns a Dirac measure to each chromatic zero of
. Under a mild hypothesis on the generating graph, we prove that the
sequence converges to some measure as tends to infinity. We
call the limiting measure of chromatic zeros associated to
. In the case of the Diamond Hierarchical Lattice we
prove that the support of has Hausdorff dimension two.
The main techniques used come from holomorphic dynamics and more specifically
the theories of activity/bifurcation currents and arithmetic dynamics. We prove
a new equidistribution theorem that can be used to relate the chromatic zeros
of a hierarchical lattice to the activity current of a particular marked point.
We expect that this equidistribution theorem will have several other
applications.Comment: To appear in Annales de l'Institut Henri Poincar\'e D. We have added
considerably more background on activity currents and especially on the
Dujardin-Favre classification of the passive locus. Exposition in the proof
of the main theorem was improved. Comments welcome
Multi-dimensional Boltzmann Sampling of Languages
This paper addresses the uniform random generation of words from a
context-free language (over an alphabet of size ), while constraining every
letter to a targeted frequency of occurrence. Our approach consists in a
multidimensional extension of Boltzmann samplers \cite{Duchon2004}. We show
that, under mostly \emph{strong-connectivity} hypotheses, our samplers return a
word of size in and exact frequency in
expected time. Moreover, if we accept tolerance
intervals of width in for the number of occurrences of each
letters, our samplers perform an approximate-size generation of words in
expected time. We illustrate these techniques on the
generation of Tetris tessellations with uniform statistics in the different
types of tetraminoes.Comment: 12p
Exclusion processes in higher dimensions: Stationary measures and convergence
There has been significant progress recently in our understanding of the
stationary measures of the exclusion process on . The corresponding
situation in higher dimensions remains largely a mystery. In this paper we give
necessary and sufficient conditions for a product measure to be stationary for
the exclusion process on an arbitrary set, and apply this result to find
examples on and on homogeneous trees in which product measures are
stationary even when they are neither homogeneous nor reversible. We then begin
the task of narrowing down the possibilities for existence of other stationary
measures for the process on . In particular, we study stationary measures
that are invariant under translations in all directions orthogonal to a fixed
nonzero vector. We then prove a number of convergence results as
for the measure of the exclusion process. Under appropriate initial conditions,
we show convergence of such measures to the above stationary measures. We also
employ hydrodynamics to provide further examples of convergence.Comment: Published at http://dx.doi.org/10.1214/009117905000000341 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Strong local survival of branching random walks is not monotone
The aim of this paper is the study of the strong local survival property for
discrete-time and continuous-time branching random walks. We study this
property by means of an infinite dimensional generating function G and a
maximum principle which, we prove, is satisfied by every fixed point of G. We
give results about the existence of a strong local survival regime and we prove
that, unlike local and global survival, in continuous time, strong local
survival is not a monotone property in the general case (though it is monotone
if the branching random walk is quasi transitive). We provide an example of an
irreducible branching random walk where the strong local property depends on
the starting site of the process. By means of other counterexamples we show
that the existence of a pure global phase is not equivalent to nonamenability
of the process, and that even an irreducible branching random walk with the
same branching law at each site may exhibit non-strong local survival. Finally
we show that the generating function of a irreducible BRW can have more than
two fixed points; this disproves a previously known result.Comment: 19 pages. The paper has been deeply reorganized and two pictures have
been added. arXiv admin note: substantial text overlap with arXiv:1104.508
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