arXiv subjects

Satoshi Handa

Publications and source records attributed to Satoshi Handa.

6 recordsLinked to original sources

arXiv2021

Mean-field behavior of nearest-neighbor oriented percolation on the BCC lattice above $8+1$ dimensions

In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the $d$-dimensional body-centered cubic (BCC) lattice $\mathbb{L}^d$ and the set of non-negative integers $\mathbb{Z}_+$. Thanks to the orderly structure of the BCC lattice, we prove that the infrared bound holds on $\mathbb{L}^d\times\mathbb{Z}_+$ in all dimensions $d\geq 9$. As opposed to ordinary percolation, we have to deal with complex numbers due to asymmetry induced by time-orientation, which makes it hard to bound the bootstrap functions in the lace-expansion analysis. By investigating the Fourier-Laplace transform of the random-walk Green function and the two-point function, we derive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yang's bound. The issue is caused by the fact that the Fourier transform of the random-walk transition probability can take the value $-1$.

math-ph↗

arXiv2021

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

Inspired by Fr\"{o}hlich-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 $\alpha>d$ and improves previous results obtained by Park for $\alpha>3d+1$, and by Ginibre, Grossmann, and Ruelle for $\alpha> d+1$, where $\alpha$ 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 $\delta>0$ as $h^*|x|^{-\delta}$, where $h^* >0$. For $d<\alpha \alpha-d$, and when $h^*$ is small enough over the critical line $\delta=\alpha-d$. For $\alpha \geq d+1$, $\delta>1$ is enough to prove the phase transition, and for $\delta=1$ 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↗

arXiv2020

Mixing time and simulated annealing for the stochastic cellular automata

Finding a ground state of a given Hamiltonian of an Ising model on a graph $G=(V,E)$ is an important but hard problem. The standard approach for this kind of problem is the application of algorithms that rely on single-spin-flip Markov chain Monte Carlo methods, such as the simulated annealing based on Glauber or Metropolis dynamics. In this paper, we investigate a particular kind of stochastic cellular automata, in which all spins are updated independently and simultaneously. We prove that (i) if the temperature is fixed sufficiently high, then the mixing time is at most of order $\log|V|$, and that (ii) if the temperature drops in time $n$ as $1/\log n$, then the limiting measure is uniformly distributed over the ground states. We also provide some simulations of the algorithms studied in this paper implemented on a GPU and show their superior performance compared to the conventional simulated annealing.

math.PR↗

arXiv2019

Finding optimal solutions by stochastic cellular automata

Finding a ground state of a given Hamiltonian is an important but hard problem. One of the potential methods is to use a Markov chain Monte Carlo (MCMC) to sample the Gibbs distribution whose highest peaks correspond to the ground states. In this short paper, we use stochastic cellular automata (SCA) and see if it is possible to find a ground state faster than the conventional MCMCs, such as the Glauber dynamics. We show that, if the temperature is sufficiently high, it is possible for SCA to have more spin-flips per update in average than Glauber and, at the same time, to have an equilibrium distribution ``close" to the one for Glauber, i.e., the Gibbs distribution. During the course, we also propose a new way to characterize how close a probability measure is to the target Gibbs.

math.OC↗

arXiv2017

A survey on the lace expansion for the nearest-neighbor models on the BCC lattice

The aim of this survey is to explain, in a self-contained and relatively beginner-friendly manner, the lace expansion for the nearest-neighbor models of self-avoiding walk and percolation that converges in all dimensions above 6 and 9, respectively. To achieve this, we consider a $d$-dimensional version of the body-centered cubic (BCC) lattice, on which it is extremely easy to enumerate various random-walk quantities. Also, we choose a particular set of bootstrapping functions, by which a notoriously complicated part of the lace-expansion analysis becomes rather transparent.

math.PR↗

arXiv2016

Mean-field bound on the 1-arm exponent for Ising ferromagnets in high dimensions

The 1-arm exponent $\rho$ for the ferromagnetic Ising model on $\mathbb{Z}^d$ is the critical exponent that describes how fast the critical 1-spin expectation at the center of the ball of radius $r$ surrounded by plus spins decays in powers of $r$. Suppose that the spin-spin coupling $J$ is translation-invariant, $\mathbb{Z}^d$-symmetric and finite-range. Using the random-current representation and assuming the anomalous dimension $\eta=0$, we show that the optimal mean-field bound $\rho\le1$ holds for all dimensions $d>4$. This significantly improves a bound previously obtained by a hyperscaling inequality.

math-ph↗