16,618 research outputs found

    The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields

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    We consider an "elastic" version of the statistical mechanical monomer-dimer problem on the n-dimensional integer lattice. Our setting includes the classical "rigid" formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan" EDM systems where the dimer potential is a weighted l1-distance and the auxiliary GRF is a Markov random field of Pickard type which behaves in space like autoregressive processes do in time. For one-dimensional Manhattan EDM systems, we compute the MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a compact transfer operator on a Hilbert space which is related to the annihilation and creation operators of the quantum harmonic oscillator and also recast it as the eigenvalue problem for a pantograph functional-differential equation.Comment: 24 pages, 4 figures, submitted on 14 October 2011 to a special issue of DCDS-

    Using Dynamical Systems to Construct Infinitely Many Primes

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    Euclid's proof can be reworked to construct infinitely many primes, in many different ways, using ideas from arithmetic dynamics. After acceptance Soundararajan noted the beautiful and fast converging formula: τ=a1/(d1)x0limnm=1n(xmaxm1d)1/dm \tau = a^{1/(d-1)} x_0 \cdot \lim_{n\to \infty} \prod_{m=1}^n \left(\frac{x_m}{ax_{m-1}^d} \right)^{1/d^m} Comment: To appear in the American Mathematical Monthl

    Discrete integrable systems generated by Hermite-Pad\'e approximants

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    We consider Hermite-Pad\'e approximants in the framework of discrete integrable systems defined on the lattice Z2\mathbb{Z}^2. We show that the concept of multiple orthogonality is intimately related to the Lax representations for the entries of the nearest neighbor recurrence relations and it thus gives rise to a discrete integrable system. We show that the converse statement is also true. More precisely, given the discrete integrable system in question there exists a perfect system of two functions, i.e., a system for which the entire table of Hermite-Pad\'e approximants exists. In addition, we give a few algorithms to find solutions of the discrete system.Comment: 20 page

    A renormalisation group method. II. Approximation by local polynomials

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    This paper is the second in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. The method is set within a normed algebra N\mathcal{N} of functionals of the fields. In this paper, we develop a general method---localisation---to approximate an element of N\mathcal{N} by a local polynomial in the fields. From the point of view of the renormalisation group, the construction of the local polynomial corresponding to FF in N\mathcal{N} amounts to the extraction of the relevant and marginal parts of FF. We prove estimates relating FF and its corresponding local polynomial, in terms of the TϕT_{\phi} semi-norm introduced in part I of the series.Comment: 30 page
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