860 research outputs found
Avoiding 2-binomial squares and cubes
Two finite words are 2-binomially equivalent if, for all words of
length at most 2, the number of occurrences of as a (scattered) subword of
is equal to the number of occurrences of in . This notion is a
refinement of the usual abelian equivalence. A 2-binomial square is a word
where and are 2-binomially equivalent.
In this paper, considering pure morphic words, we prove that 2-binomial
squares (resp. cubes) are avoidable over a 3-letter (resp. 2-letter) alphabet.
The sizes of the alphabets are optimal
Relations on words
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.
In the second part, we mainly focus on abelian equivalence, -abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and -equivalence. In particular, some new refinements of abelian equivalence are introduced
On the Mixing Time of Geographical Threshold Graphs
We study the mixing time of random graphs in the -dimensional toric unit
cube generated by the geographical threshold graph (GTG) model, a
generalization of random geometric graphs (RGG). In a GTG, nodes are
distributed in a Euclidean space, and edges are assigned according to a
threshold function involving the distance between nodes as well as randomly
chosen node weights, drawn from some distribution. The connectivity threshold
for GTGs is comparable to that of RGGs, essentially corresponding to a
connectivity radius of . However, the degree distributions
at this threshold are quite different: in an RGG the degrees are essentially
uniform, while RGGs have heterogeneous degrees that depend upon the weight
distribution. Herein, we study the mixing times of random walks on
-dimensional GTGs near the connectivity threshold for . If the
weight distribution function decays with for an arbitrarily small constant then the mixing time
of GTG is \mixbound. This matches the known mixing bounds for the
-dimensional RGG
Correlation Functions of XX0 Heisenberg Chain, q-Binomial Determinants, and Random Walks
The XX0 Heisenberg model on a cyclic chain is considered. The representation
of the Bethe wave functions via the Schur functions allows to apply the
well-developed theory of the symmetric functions to the calculation of the
thermal correlation functions. The determinantal expressions of the
form-factors and of the thermal correlation functions are obtained. The
q-binomial determinants enable the connection of the form-factors with the
generating functions both of boxed plane partitions and of self-avoiding
lattice paths. The asymptotical behavior of the thermal correlation functions
is studied in the limit of low temperature provided that the characteristic
parameters of the system are large enough.Comment: 27 pages, 2 figures, LaTe
Threshold functions and Poisson convergence for systems of equations in random sets
We present a unified framework to study threshold functions for the existence
of solutions to linear systems of equations in random sets which includes
arithmetic progressions, sum-free sets, -sets and Hilbert cubes. In
particular, we show that there exists a threshold function for the property
" contains a non-trivial solution of
", where is a random set and each of
its elements is chosen independently with the same probability from the
interval of integers . Our study contains a formal definition of
trivial solutions for any combinatorial structure, extending a previous
definition by Ruzsa when dealing with a single equation.
Furthermore, we study the behaviour of the distribution of the number of
non-trivial solutions at the threshold scale. We show that it converges to a
Poisson distribution whose parameter depends on the volumes of certain convex
polytopes arising from the linear system under study as well as the symmetry
inherent in the structures, which we formally define and characterize.Comment: New version with minor corrections and changes in notation. 24 Page
Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions
We introduce spatially explicit stochastic processes to model multispecies
host-symbiont interactions. The host environment is static, modeled by the
infinite percolation cluster of site percolation. Symbionts evolve on the
infinite cluster through contact or voter type interactions, where each host
may be infected by a colony of symbionts. In the presence of a single symbiont
species, the condition for invasion as a function of the density of the habitat
of hosts and the maximal size of the colonies is investigated in details. In
the presence of multiple symbiont species, it is proved that the community of
symbionts clusters in two dimensions whereas symbiont species may coexist in
higher dimensions.Comment: Published in at http://dx.doi.org/10.1214/10-AAP734 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nature of the collapse transition in interacting self-avoiding trails
We study the interacting self-avoiding trail (ISAT) model on a Bethe lattice
of general coordination and on a Husimi lattice built with squares and
coordination . The exact grand-canonical solutions of the model are
obtained, considering that up to monomers can be placed on a site and
associating a weight for a -fold visited site. Very rich phase
diagrams are found with non-polymerized (NP), regular polymerized (P) and dense
polymerized (DP) phases separated by lines (or surfaces) of continuous and
discontinuous transitions. For Bethe lattice with and , the collapse
transition is identified with a bicritical point and the collapsed phase is
associated to the dense polymerized phase (solid-like) instead of the regular
polymerized phase (liquid-like). A similar result is found for the Husimi
lattice, which may explain the difference between the collapse transition for
ISAT's and for interacting self-avoiding walks on the square lattice. For
and (studied on the Bethe lattice only), a more complex phase diagram is
found, with two critical planes and two coexistence surfaces, separated by two
tricritical and two critical end-point lines meeting at a multicritical point.
The mapping of the phase diagrams in the canonical ensemble is discussed and
compared with simulational results for regular lattices.Comment: 12 pages, 13 figure
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