We compute rigorously the scaling limit of multi-point energy correlations in the critical Ising model on a torus. For the one-point function, averaged between horizontal and vertical edges of the square lattice, this result has been known since the 1969 work of Ferdinand and Fischer. We propose an alternative proof, in a slightly greater generality, via a new exact formula in terms of determinants of discrete Laplacians. We also compute the main term of the asymptotics of the difference $\mathbb{E}(ε_{V}-ε_{H})$ of the energy density on a vertical and a horizontal edge, which is of order of $δ^{2}$, where $δ$ is the mesh size. The observable $ε_{V}-ε_{H}$ has been identified by Kadanoff and Ceva as (a component of) the stress-energy tensor.
We then apply the discrete complex analysis methods of Smirnov and Hongler to compute the multi-point correlations. The fermionic observables are only periodic with doubled periods; by anti-symmetrization, this leads to contributions from four "sectors". The main new challenge arises in the doubly periodic sector, due to the existence of non-zero constant (discrete) analytic functions. We show that some additional input, namely the scaling limit of the one-point function and of relative contribution of sectors to the partition function, is sufficient to overcome this difficulty and successfully compute all correlations.
We present detailed calculations for the partition function and the free energy of the finite two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions, variable aspect ratio, and anisotropic couplings, as well as for the corresponding universal free energy finite-size scaling functions. Therefore, we review the dimer mapping, as well as the interplay between its topology and the different types of boundary conditions. As a central result, we show how both the finite system as well as the scaling form decay into contributions for the bulk, a characteristic f…
An extension of the Ising spin configurations to continuous functions is used for an exact representation of the Random Field Ising Model's order parameter in terms of disagreement percolation. This facilitates an extension of the recent analyses of the decay of correlations to positive temperatures, at homogeneous but arbitrarily weak disorder.
We study ferromagnetic Ising models on finite graphs with an inhomogeneous external field, where a subset of vertices is designated as the boundary. We show that the influence of boundary conditions on any given spin is maximised when the external field is identically $0$. One corollary is that spin-spin correlations are maximised when the external field vanishes and the boundary condition is free, which proves a conjecture of Shlosman. In particular, the random field Ising model on ${\mathbb Z}^d$, $d\geq 3$, exhibits exponential decay of correlations in the entire high temperature regime of …
We prove convergence of renormalized correlations of primary fields, i. e., spins, disorders, fermions and energy densities, in the scaling limit of the critical Ising model in arbitrary finitely connected domains, with fixed (plus or minus) or free boundary conditions, or mixture thereof. We describe the limits of correlations in terms of solutions of Riemann boundary value problems, and prove their conformal covariance. Moreover, we prove fusion rules, or operator product expansions, which describe asymptotics of the scaling limits of the correlations as some of the points collide together. …
For the two-dimensional random field Ising model where the random field is given by i.i.d.\ mean zero Gaussian variables with variance $ε^2$, we study (one natural notion of) the correlation length, which is the critical size of a box at which the influences of the random field and of the boundary condition on the spin magnetization are comparable. We show that as $ε\to 0$, at zero temperature the correlation length scales as $e^{Θ(ε^{-4/3})}$ (and our upper bound applies for all positive temperatures).
Given a finite rhombus tiling of a polygonal region in the plane, the associated critical $Z$-invariant Ising model is invariant under star-triangle transformations. We give a simple matrix formula describing spin correlations between boundary vertices in terms of the shape of the region. When the region is a regular polygon, our formula becomes an explicit trigonometric sum.
We consider finite-range ferromagnetic Ising models on $\mathbb{Z}^d$ in the regime $β<β_c$. We analyze the behavior of the prefactor to the exponential decay of $\mathrm{Cov}(σ_A,σ_B)$, for arbitrary finite sets $A$ and $B$ of even cardinality, as the distance between $A$ and $B$ diverges.
We consider a measure given as the continuum limit of a one-dimensional Ising model with long-range translationally invariant interactions. Mathematically, the measure can be described by a self-interacting Poisson driven jump process. We prove a correlation inequality, estimating the magnetic susceptibility of this model, which holds for small $L^1$-norm of the interaction function. The bound on the magnetic susceptibility has applications in quantum field theory and can be used to prove existence of ground states for the spin boson model.
We prove the analogue of the strong Szeg{\H o} limit theorem for a large class of bordered Toeplitz determinants. In particular, by applying our results to the formula of Au-Yang and Perk \cite{YP} for the next-to-diagonal correlations $\langle σ_{0,0}σ_{N-1,N} \rangle$ in the anisotropic square lattice Ising model, we rigorously justify that the next-to-diagonal long-range order is the same as the diagonal and horizontal ones in the low temperature regime. The anisotropy-dependence of the subleading term in the asymptotics of the next-to-diagonal correlations is also established. We use Riema…
Correlation inequalities have played an essential role in the analysis of ferromagnetic models but have not been established in spin glass models. In this study, we obtain some correlation inequalities for the Ising models with quenched randomness, where the distribution of the interactions is symmetric. The acquired inequalities can be regarded as an extension of the previous results, which were limited to the local energy for a spin set, to the local energy for a pair of spin sets. Besides, we also obtain some correlation inequalities for asymmetric distribution.
