Related papers: Power series expansions for the planar monomer-dim…
We compute the high-dimensional limit of the free energy associated with a multi-layer generalized linear model. Under certain technical assumptions, we identify the limit in terms of a variational formula. The approach is to first show…
By using the asymptotic theory of Pemantle and Wilson, exact asymptotic expansions of the free energy of the monomer-dimer model on rectangular $n \times \infty$ lattices in terms of dimer density are obtained for small values of $n$, at…
Recently an expansion as a power series in 1/d has been presented for the specific entropy of a complete dimer covering of a d-dimensional hypercubic lattice. This paper extends from 3 to 10 the number of terms known in the series. Likewise…
We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation a fundamental domain and linear algebra.
In this work we obtain the planar free energy for the Hermitian one-matrix model with various choices of the potential. We accomplish this by applying an approach that bypasses the usual diagonalization of the matrices and the introduction…
The free energy of the Penner model is shown to be closely related to the integral over the two diagonalizing unitary matrices of a complex rectangular matrix.
This work addresses the question of whether it is possible to define simple pair-wise interaction terms to approximate free energies of proteins or polymers. Rather than ask how reliable a potential of mean force is, one can ask how…
A new computational strategy is proposed to approximate, with a simple but accurate expression, the Maier- Saupe free energy for nematic order. Instead of the traditional approach of expanding the free energy with a truncated Taylor series,…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
We have substantially extended the high-temperature and low-magnetic-field (and the related low-temperature and high-magnetic-field) bivariate expansions of the free energy for the conventional three-dimensional Ising model and for a…
Systems with many interacting stochastic constituents are fully characterized by their free energy. Computing this quantity is therefore the objective of various approaches, notably perturbative expansions, which are applied in problems…
We outline the most recent theory for the computation of the exponential growth rate of the number of configurations on a multi-dimensional grid. As an application we compute the monomer-dimer constant for the 2-dimensional grid to 8…
The expression of the free energy density of a classical crystalline system as a gradient expansion in terms of a set of order parameters is developed using classical density functional theory. The goal here is to extend and complete an…
A mathematical framework is constructed for the sum of the lowest N eigenvalues of a potential. Exactness is illustrated on several model systems (harmonic oscillator, particle in a box, and Poschl-Teller well). Its order-by-order…
We study low-energy expansion and high-energy expansion of reflection coefficients for one-dimensional Schr\"odinger equation, from which expansions of the Green function can be obtained. Making use of the equivalent Fokker-Planck equation,…
I review my new method for solving general 1-matrix models by expanding in $N^{-1}$ without taking a physical continuum limit. Using my method, each coefficient of the free energy in the genus expansion is exactly computable. One can…
We consider a binary system of small and large spheres of finite size in a continuous medium interacting via a non-negative potential. We work in the canonical ensemble and compute upper and lower bound for the free energy at finite and…
We consider dimer model on a hexagonal lattice. This model can be seen as a "pile of cubes in the box". The energy of configuration is given by the volume of the pile and the partition function is computed by the classical MacMahon formula…
We generalize a recently proposed small-energy expansion for one-dimensional quantum-mechanical models. The original approach was devised to treat symmetric potentials and here we show how to extend it to non-symmetric ones. Present…
We develop a method for computing series expansions for solutions to ordinary differential equations when the asymptotic form contains both linear and logarithmic terms. Such situations are common in primordial cosmology when considering…