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Rafael Leon Greenblatt

Publications and source records attributed to Rafael Leon Greenblatt.

6 recordsLinked to original sources
arXiv2025

The scaling limit of boundary spin correlations in non-integrable Ising models

We consider a class of non-integrable 2D Ising models obtained by perturbing the nearest-neighbor model via a weak, finite range potential which preserves translation and spin-flip symmetry, and we study its critical theory in the half-plane. We prove that the leading order long-distance behavior of the correlation functions for spins on the boundary is the same as for the nearest-neighbor model, up to an analytic multiplicative renormalization constant. In particular, the scaling limit is the Pfaffian of an explicit matrix. The proof is based on an exact representation of the generating function of correlations in terms of a Grassmann integral and on a multiscale analysis thereof, generalizing previous results to include observables located on the boundary.

math-ph↗
arXiv2025

The boundary disorder correlation for the Ising model on a cylinder

I give an expression for the correlation function of disorder insertions on the edges of the critical Ising model on a cylinder as a function of the aspect ratio (rescaled in the case of anisotropic couplings). This is obtained from an expression for the finite size scaling term in the free energy on a cylinder in periodic and antiperiodic boundary conditions in terms of Jacobi theta functions.

cond-mat.stat-mech↗
arXiv2023

Discrete and zeta-regularized determinants of the Laplacian on polygonal domains with Dirichlet boundary conditions

For $Π\subset \mathbb{R}^2$ a connected, open, bounded set whose boundary is a finite union of disjoint polygons whose vertices have integer coordinates, the logarithm of the discrete Laplacian on $LΠ\cap \mathbb{Z}^2$ with Dirichlet boundary conditions has an asymptotic expansion for large $L$ involving the zeta-regularized determinant of the associated continuum Laplacian. When $Π$ is not simply connected, this result extends to Laplacians acting on two-valued functions with a specified monodromy class.

math-ph↗
arXiv2022

Energy correlations of non-integrable Ising models: The scaling limit in the cylinder

We consider a class of non-integrable 2D Ising models, whose Hamiltonian, in addition to the nearest neighbor couplings, includes weak multi-spin interactions, even under spin flip. We study the model in cylindrical domains of arbitrary aspect ratio and prove that, in the scaling limit, the multipoint energy correlations converge to the same limiting correlations as those of the nearest-neighbor Ising model in the cylinder with renormalized couplings, up to an overall multiplicative constant, independent of the shape and size of the domain. The proof is based on a representation of the generating function of correlations in terms of a non-Gaussian Grassmann integral, and a constructive Renormalization Group (RG) analysis thereof. A key technical novelty compared with previous works is a systematic analysis of the effect of the boundary corrections to the RG flow, in particular a proof that the scaling dimension of boundary operators is better by one dimension than their bulk counterparts. A cancellation mechanism based on an approximate image rule for the fermionic Green's function is of crucial importance for controlling the RG flow of the marginal boundary terms.

math-ph↗
arXiv2016

Singular behavior of the leading Lyapunov exponent of a product of random $2 \times 2$ matrices

We consider a certain infinite product of random $2 \times 2$ matrices appearing in the solution of some $1$ and $1+1$ dimensional disordered models in statistical mechanics, which depends on a parameter $\varepsilon>0$ and on a real random variable with distribution $μ$. For a large class of $μ$, we prove the prediction by B. Derrida and H. J. Hilhorst (J. Phys. A 16:2641, 1983) that the Lyapunov exponent behaves like $C \varepsilon^{2 α}$ in the limit $\varepsilon \searrow 0$, where $α\in (0,1)$ and $C>0$ are determined by $μ$. Derrida and Hilhorst performed a two-scale analysis of the integral equation for the invariant distribution of the Markov chain associated to the matrix product and obtained a probability measure that is expected to be close to the invariant one for small $\varepsilon$. We introduce suitable norms and exploit contractivity properties to show that such a probability measure is indeed close to the invariant one in a sense which implies a suitable control of the Lyapunov exponent.

math-ph↗