860 research outputs found

    Avoiding 2-binomial squares and cubes

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    Two finite words u,vu,v are 2-binomially equivalent if, for all words xx of length at most 2, the number of occurrences of xx as a (scattered) subword of uu is equal to the number of occurrences of xx in vv. This notion is a refinement of the usual abelian equivalence. A 2-binomial square is a word uvuv where uu and vv 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

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    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, kk-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and MM-equivalence. In particular, some new refinements of abelian equivalence are introduced

    On the Mixing Time of Geographical Threshold Graphs

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    We study the mixing time of random graphs in the dd-dimensional toric unit cube [0,1]d[0,1]^d 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 r=(logn/n)1/dr=(\log n/n)^{1/d}. 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 dd-dimensional GTGs near the connectivity threshold for d2d \geq 2. If the weight distribution function decays with P[Wx]=O(1/xd+ν)\mathbb{P}[W \geq x] = O(1/x^{d+\nu}) for an arbitrarily small constant ν>0\nu>0 then the mixing time of GTG is \mixbound. This matches the known mixing bounds for the dd-dimensional RGG

    Correlation Functions of XX0 Heisenberg Chain, q-Binomial Determinants, and Random Walks

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    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

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    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, Bh[g]B_{h}[g]-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property "A\mathcal{A} contains a non-trivial solution of Mx=0M\cdot\textbf{x}=\textbf{0}", where A\mathcal{A} is a random set and each of its elements is chosen independently with the same probability from the interval of integers {1,,n}\{1,\dots,n\}. 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

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    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

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    We study the interacting self-avoiding trail (ISAT) model on a Bethe lattice of general coordination qq and on a Husimi lattice built with squares and coordination q=4q=4. The exact grand-canonical solutions of the model are obtained, considering that up to KK monomers can be placed on a site and associating a weight ωi\omega_i for a ii-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 q=4q=4 and K=2K=2, 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 q=6q=6 and K=3K=3 (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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