Related papers: Fredholm Determinant evaluations of the Ising Mode…
We recall the form factors $ f^{(j)}_{N,N}$ corresponding to the $\lambda$-extension $C(N,N; \lambda)$ of the two-point diagonal correlation function of the Ising model on the square lattice and their associated linear differential…
We consider a two-dimensional Ising model with random i.i.d. nearest-neighbor ferromagnetic couplings and no external magnetic field. We show that, if the probability of supercritical couplings is small enough, the system admits a…
In a recent contribution, Dotsenko establishes a Fredholm determinant formula for the two-point distribution of the KPZ equation in the long time limit and starting from narrow wedge initial conditions. We establish that his expression is…
Using the variational formula for operator product coefficients a method for perturbative calculation of the short-distance expansion of the Spin-Spin correlation function in the two dimensional Ising model is presented. Results of explicit…
Ilse Fischer and the second author introduced in [Algebr. Comb. 7 (2024), no. 5, 1319-1345] a two parameter family of polynomials defined as sums over totally symmetric plane partitions and connected to alternating sign matrices and…
We consider a completely integrable lattice regularization of the sine-Gordon model with discrete space and continuous time. We derive a determinant representation for a correlation function which in the continuum limit turns into the…
Two multiplicative anomalies are evaluated for the determinant of the conformal higher spin propagating operator on spheres given by Tseytlin. One holds for the decomposition of the higher derivative product into its individual second order…
We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable…
We show that the ratio of a discrete Toeplitz/Hankel determinant and its continuous counterpart equals a Freholm determinant involving continuous orthogonal polynomials. This identity is used to evaluate a triple asymptotic of some discrete…
Using exact expressions for the Ising form factors, we give a new very simple proof that the spin-spin and disorder-disorder correlation functions are governed by the Painlev\'e III non linear differential equation. We also show that the…
Parasupersymmetric quantum mechanics is exploited to introduce a topological invariant associated with a pair of parameter dependent Fredholm (respectively elliptic differential) operators satisfying two compatibility conditions. An…
We consider a family of linear operators, diagonalized by the Hankel transform. The Fredholm determinants of these operators, restricted to $L_2[0, R]$, are expressed in a convenient form for asymptotic analysis as $R\to\infty$. The result…
The spontaneous magnetization of a two-dimensional lattice model can be expressed in terms of the partition function $W$ of a system with fixed boundary spins and an extra weight dependent on the value of a particular central spin. For the…
We study the Fredholm properties of a general class of elliptic differential operators on $\R^n$. These results are expressed in terms of a class of weighted function spaces, which can be locally modeled on a wide variety of standard…
The parameter imbedding method for the Fredholm equation converts it into an initial value problem in its parameter $\lambda$. We establish the method for general operator equations of the form $[I + \hat{f}(\lambda)]\psi = \phi $. It is…
The Peierls argument is a mathematically rigorous and intuitive method to show the presence of a non-vanishing spontaneous magnetization in some lattice models. This argument is typically explained for the $D=2$ Ising model in a way which…
We review the connections between the octahedral recurrence, $\lambda$-determinants and tiling problems. This provides in particular a direct combinatorial interpretation of the $\lambda$-determinant (and generalizations thereof) of an…
The six-vertex model with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is obtained. The kernel of the corrtesponding integral operator depends on Laguerre…
We revisit the problem of finding the probability distribution of a fermionic number of one-dimensional spinless free fermions on a segment of a given length. The generating function for this probability distribution can be expressed as a…
Ashkin-Teller model is a two-layer lattice model where spins in each layer interact ferromagnetically with strength $J$, and the spin-dipoles (product of spins) interact with neighbors with strength $\lambda.$ The model exhibits…