arXiv subjects

Eric O. Endo

Publications and source records attributed to Eric O. Endo.

7 recordsLinked to original sources

arXiv2024

Long-Range Ising Models: Contours, Phase Transitions and Decaying Fields

Inspired by Fröhlich-Spencer and subsequent authors who introduced the notion of contour for long-range systems, we provide a definition of contour and a direct proof for the phase transition for ferromagnetic long-range Ising models on $\mathbb{Z}^d$, $d\geq 2$. The argument, which is based on a multi-scale analysis, works for the sharp region $α>d$ and improves previous results obtained by Park for $α>3d+1$, and by Ginibre, Grossmann, and Ruelle for $α> d+1$, where $α$ is the power of the coupling constant. The key idea is to avoid a large number of small contours. As an application, we prove the persistence of the phase transition when we add a polynomially decaying magnetic field with power $δ>0$ as $h^*|x|^{-δ}$, where $h^* >0$. For $d<α α-d$, and when $h^*$ is small enough over the critical line $δ=α-d$. For $α\geq d+1$, $δ>1$ is enough to prove the phase transition, and for $δ=1$ we have to ask $h^*$ small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field.

math-ph↗

arXiv2024

Local Central Limit Theorem for unbounded long-range potentials

We prove the equivalence between the integral central limit theorem and the local central limit theorem for two-body potentials with long-range interactions on the lattice $\mathbb{Z}^d$ for $d\ge 1$. The spin space can be an arbitrary, possibly unbounded subset of the real axis with a suitable a-priori measure. For general unbounded spins, our method works at high-enough temperature, but for bounded spins our results hold for every temperature. Our proof relies on the control of the integrated characteristic function, which is achieved by dividing the integration into three different regions, following a standard approach proposed forty years ago by Campanino, Del Grosso and Tirozzi. The bounds required in the different regions are obtained through cluster-expansion techniques. For bounded spins, the arbitrariness of the temperature is achieved through a decimation ("dilution") technique, also introduced in the later reference.

math-ph↗

arXiv2024

On Long Range Ising Models with Random Boundary Conditions

We consider polynomial long-range Ising models in one dimension, with ferromagnetic pair interactions decaying with power $2-α$ (for $0 \leq α< 1$), and prepared with randomly chosen boundary conditions. We show that at low temperatures in the thermodynamic limit the finite-volume Gibbs measures do not converge, but have a distributional limit, the so-called metastate. We find that there is a distinction between the values of $α$ less than or larger than $\frac{1}{2}$. For moderate, or intermediate, decay $α< \frac{1}{2}$, the metastate is very dispersed and supported on the set of all Gibbs measures, both extremal and non-extremal, whereas for slow decays $α> \frac{1}{2}$ the metastate is still dispersed, but has its support just on the set of the two extremal Gibbs measures, the plus measure and the minus measure. The former, moderate decays case, appears to be new and is due to the occurrence of almost sure boundedness of the random variable which is the sum of all interaction (free) energies between random and ordered half-lines, when the decay is fast enough, but still slow enough to get a phase transition ($α>0$); while the latter, slow decays case, is more reminiscent of and similar to the behaviour of higher-dimensional nearest-neighbour Ising models with diverging boundary (free) energies. We leave the threshold case $α=\frac{1}{2}$ for further studies.

math-ph↗

arXiv2022

Local Central Limit Theorem for Long-Range Two-Body Potentials at Sufficiently High Temperatures

Dobrushin and Tirozzi [14] showed that, for a Gibbs measure with the finite-range potential, the Local Central Limit Theorem is implied by the Integral Central Limit Theorem. Campanino, Capocaccia, and Tirozzi [7] extended this result for a family of Gibbs measures for long-range pair potentials satisfying certain conditions. We are able to show for a family of Gibbs measures for long-range pair potentials not satisfying the conditions given in [7], that at sufficiently high temperatures, if the Integral Central Limit Theorem holds for a given sequence of Gibbs measures, then the Local Central Limit Theorem also holds for the same sequence. We also extend [7] when the state space is general, provided that it is equipped with a finite measure.

math-ph↗

arXiv2020

Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts

We consider the natural definition of DLR measure in the setting of $σ$-finite measures on countable Markov shifts. We prove that the set of DLR measures contains the set of conformal measures associated with Walters potentials. In the BIP case, or when the potential normalizes the Ruelle's operator, we prove that the notions of DLR and conformal coincide. On the standard renewal shift, we study the problem of describing the cases when the set of the eigenmeasures jumps from finite to infinite measures when we consider high and low temperatures, respectively. For this particular shift, we prove that there always exist finite DLR measures, and we have an expression to the critical temperature for this volume-type phase transition, which occurs only for potentials with the infinite first variation.

math.DS↗

arXiv2019

The roles of random boundary conditions in spin systems

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these results.

math-ph↗

arXiv2018

Contour methods for long-range Ising models: weakening nearest-neighbor interactions and adding decaying fields

We consider ferromagnetic long-range Ising models which display phase transitions. They are long-range one-dimensional Ising ferromagnets, in which the interaction is given by $J_{x,y} = J(|x-y|)\equiv \frac{1}{|x-y|^{2-α}}$ with $α\in [0, 1)$, in particular, $J(1)=1$. For this class of models one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fröhlich-Spencer contours for $α\neq 0$, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fröhlich and Spencer for $α=0$ and conjectured by Cassandro et al for the region they could treat, $α\in (0,α_{+})$ for $α_+=\log(3)/\log(2)-1$, although in the literature dealing with contour methods for these models it is generally assumed that $J(1)\gg1$, we can show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any $α\in [0,1)$. Moreover, we show that when we add a magnetic field decaying to zero, given by $h_x= h_*\cdot(1+|x|)^{-γ}$ and $γ>\max\{1-α, 1-α^* \}$ where $α^*\approx 0.2714$, the transition still persists.

math-ph↗