16,620 research outputs found
The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields
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-
Discrete integrable systems generated by Hermite-Pad\'e approximants
We consider Hermite-Pad\'e approximants in the framework of discrete
integrable systems defined on the lattice . 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
Using Dynamical Systems to Construct Infinitely Many Primes
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: Comment: To appear in the American Mathematical Monthl
A renormalisation group method. II. Approximation by local polynomials
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
of functionals of the fields. In this paper, we develop a general
method---localisation---to approximate an element of by a local
polynomial in the fields. From the point of view of the renormalisation group,
the construction of the local polynomial corresponding to in
amounts to the extraction of the relevant and marginal parts of . We prove
estimates relating and its corresponding local polynomial, in terms of the
semi-norm introduced in part I of the series.Comment: 30 page
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