Searcharxiv⌕ Search

arXiv subjects

Rafael L. Greenblatt

Publications and source records attributed to Rafael L. Greenblatt.

12 recordsLinked to original sources
arXiv2025

On the Lyapunov exponent for the random field Ising transfer matrix, in the critical case

We study the top Lyapunov exponent of a product of random $2 \times 2$ matrices appearing in the analysis of several statistical mechanical models with disorder, extending a previous treatment of the critical case (Giacomin and Greenblatt, ALEA 19 (2022), 701-728) by significantly weakening the assumptions on the disorder distribution. The argument we give completely revisits and improves the previous proof. As a key novelty we build a probability that is close to the Furstenberg probability, i.e. the invariant probability of the Markov chain corresponding to the evolution of the direction of a vector in ${\mathbb R}^2$ under the action of the random matrices, in terms of the ladder times of a centered random walk which is directly related to the random matrix sequence. We then show that sharp estimates on the ladder times (renewal) process lead to a sharp control on the probability measure we build and, in turn, to the control of its distance from the Furstenberg probability.

math.PR↗
arXiv2024

On the spin interface distribution for non-integrable variants of the two-dimensional Ising model

We point out that the construction of a martingale observable describing the spin interface of the two-dimensional Ising model extends to a class of non-integrable variants of the two-dimensional Ising model, and express it in terms of Grassmann integrals. Under a conjecture about the scaling limit of this object, which is similar to some results recently obtained using constructive renormalization group methods, this would imply that the distribution of the interface at criticality has the same scaling limit as in the integrable model: Schramm-Loewner evolution SLE(3).

math-ph↗
arXiv2023

Adiabatic Evolution of Low-Temperature Many-Body Systems

We consider finite-range, many-body fermionic lattice models and we study the evolution of their thermal equilibrium state after introducing a weak and slowly varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. Under a spectral gap assumption on the unperturbed Hamiltonian, convergence is also uniform in temperature. As an application, our expansion allows us to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. As a corollary, we also establish the validity of linear response. Our strategy is based on a rigorous version of the Wick rotation, which allows us to represent the Duhamel expansion for the real-time dynamics in terms of Euclidean correlation functions, for which precise decay estimates are proved using fermionic cluster expansion.

math-ph↗
arXiv2022

The zeros of the partition function of the pinning model

We aim at understanding for which (complex) values of the potential the pinning partition function vanishes. The pinning model is a Gibbs measure based on discrete renewal processes with power law inter-arrival distributions. We obtain some results for rather general inter-arrival laws, but we achieve a substantially more complete understanding for a specific one parameter family of inter-arrivals. We show, for such a specific family, that the zeros asymptotically lie on (and densely fill) a closed curve that, unsurprisingly, touches the real axis only in one point (the critical point of the model). We also perform a sharper analysis of the zeros close to the critical point and we exploit this analysis to approach the challenging problem of Griffiths singularities for the disordered pinning model. The techniques we exploit are both probabilistic and analytical. Regarding the first, a central role is played by limit theorems for heavy tail random variables. As for the second, potential theory and singularity analysis of generating functions, along with their interplay, will be at the heart of several of our arguments.

math-ph↗
arXiv2022

Lyapunov exponent for products of random Ising transfer matrices: the balanced disorder case

We analyze the top Lyapunov exponent of the product of sequences of two by two matrices that appears in the analysis of several statistical mechanics models with disorder: for example these matrices are the transfer matrices for the nearest neighbor Ising chain with random external field, and the free energy density of this Ising chain is the Lyapunov exponent we consider. We obtain the sharp behavior of this exponent in the large interaction limit when the external field is centered: this balanced case turns out to be critical in many respects. From a mathematical standpoint we precisely identify the behavior of the top Lyapunov exponent of a product of two dimensional random matrices close to a diagonal random matrix for which top and bottom Lyapunov exponents coincide. In particular, the Lyapunov exponent is only $\log$-Hölder continuous.

math.PR↗
arXiv2021

Non-integrable Ising models in cylindrical geometry: Grassmann representation and infinite volume limit

In this paper, meant as a companion to arXiv:2006.04458, we consider a class of non-integrable $2D$ Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green's function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries.

math-ph↗
arXiv2018

Continuum limit of random matrix products in statistical mechanics of disordered systems

We consider a particular weak disorder limit ("continuum limit") of matrix products that arise in the analysis of disordered statistical mechanics systems, with a particular focus on random transfer matrices. The limit system is a diffusion model for which the leading Lyapunov exponent can be expressed explicitly in terms of modified Bessel functions, a formula that appears in the physical literature on these disordered systems. We provide an analysis of the diffusion system as well as of the link with the matrix products. We then apply the results to the framework considered by Derrida and Hilhorst [J. Phys. A (1983)], which deals in particular with the strong interaction limit for disordered Ising model in one dimension and that identifies a singular behavior of the Lyapunov exponent (of the transfer matrix), and to the two dimensional Ising model with columnar disorder (McCoy-Wu model). We show that the continuum limit sharply captures the Derrida and Hilhorst singularity. Moreover we revisit the analysis by McCoy and Wu [Phys. Rev. 1968] and remark that it can be interpreted in terms of the continuum limit approximation. We provide a mathematical analysis of the continuum approximation of the free energy of the McCoy-Wu model, clarifying the prediction (by McCoy and Wu) that, in this approximation, the free energy of the two dimensional Ising model with columnar disorder is $C^\infty$ but not analytic at the critical temperature.

math-ph↗
arXiv2012

The scaling limit of the energy correlations in non integrable Ising models

We obtain an explicit expression for the multipoint energy correlations of a non solvable two-dimensional Ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength $λ$, in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one. Our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory, which generalizes the one originally used by Schultz, Mattis and Lieb for the nearest neighbor Ising model. The interacting model is then analyzed by a multiscale method first proposed by Pinson and Spencer. If the lattice spacing is finite, then the correlations cannot be computed in closed form: rather, they are expressed in terms of infinite, convergent, power series in $λ$. In the scaling limit, these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable Ising model, up to a finite renormalization of the parameters. Explicit bounds on the speed of convergence to the scaling limit are derived.

math-ph↗
arXiv2011

Proof of Rounding by Quenched Disorder of First Order Transitions in Low-Dimensional Quantum Systems

We prove that for quantum lattice systems in d<=2 dimensions the addition of quenched disorder rounds any first order phase transition in the corresponding conjugate order parameter, both at positive temperatures and at T=0. For systems with continuous symmetry the statement extends up to d<=4 dimensions. This establishes for quantum systems the existence of the Imry-Ma phenomenon which for classical systems was proven by Aizenman and Wehr. The extension of the proof to quantum systems is achieved by carrying out the analysis at the level of thermodynamic quantities rather than equilibrium states.

cond-mat.stat-mech↗
arXiv2010

Effects of Quenched Randomness on Classical and Quantum Phase Transitions

This dissertation describes the effect of quenched randomness on first order phase transitions in lattice systems, classical and quantum. It is proven that a large class of quantum lattice systems in low dimension (d <= 2 or, with suitable continuous symmetry, d <= 4) cannot exhibit first-order phase transitions in the presence of suitable ("direct") quenched disorder.

cond-mat.stat-mech↗
arXiv2009

Rounding of First Order Transitions in Low-Dimensional Quantum Systems with Quenched Disorder

We prove that the addition of an arbitrarily small random perturbation of a suitable type to a quantum spin system rounds a first order phase transition in the conjugate order parameter in d <= 2 dimensions, or in systems with continuous symmetry in d <= 4. This establishes rigorously for quantum systems the existence of the Imry-Ma phenomenon, which for classical systems was proven by Aizenman and Wehr.

cond-mat.stat-mech↗