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Akira Sakai

Publications and source records attributed to Akira Sakai.

At least 37 records · Page 2Linked to original sources
arXiv2021

Bermuda Triangles: GNNs Fail to Detect Simple Topological Structures

Most graph neural network architectures work by message-passing node vector embeddings over the adjacency matrix, and it is assumed that they capture graph topology by doing that. We design two synthetic tasks, focusing purely on topological problems -- triangle detection and clique distance -- on which graph neural networks perform surprisingly badly, failing to detect those "bermuda" triangles. Datasets and their generation scripts are publicly available on github.com/FujitsuLaboratories/bermudatriangles and dataset.labs.fujitsu.com.

cs.LG↗
arXiv2019

Crossover phenomena in the critical behavior for long-range models with power-law couplings

This is a short review of the two papers on the $x$-space asymptotics of the critical two-point function $G_{p_c}(x)$ for the long-range models of self-avoiding walk, percolation and the Ising model on $\mathbb{Z}^d$, defined by the translation-invariant power-law step-distribution/coupling $D(x)\propto|x|^{-d-α}$ for some $α>0$. Let $S_1(x)$ be the random-walk Green function generated by $D$. We have shown that $\bullet~~S_1(x)$ changes its asymptotic behavior from Newton ($α>2$) to Riesz ($α<2$), with log correction at $α=2$; $\bullet~~G_{p_c}(x)\sim\frac{A}{p_c}S_1(x)$ as $|x|\to\infty$ in dimensions higher than (or equal to, if $α=2$) the upper critical dimension $d_c$ (with sufficiently large spread-out parameter $L$). The model-dependent $A$ and $d_c$ exhibit crossover at $α=2$. The keys to the proof are (i) detailed analysis on the underlying random walk to derive sharp asymptotics of $S_1$, (ii) bounds on convolutions of power functions (with log corrections, if $α=2$) to optimally control the lace-expansion coefficients $π_p^{(n)}$, and (iii) probabilistic interpretation (valid only when $α\le2$) of the convolution of $D$ and a function $\varPi_p$ of the alternating series $\sum_{n=0}^\infty(-1)^nπ_p^{(n)}$. We outline the proof, emphasizing the above key elements for percolation in particular.

math-ph↗
arXiv2019

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

The 1-arm exponent $ρ$ 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 $η=0$, we show that the optimal mean-field bound $ρ\le1$ holds for all dimensions $d>4$. This significantly improves a bound previously obtained by a hyperscaling inequality.

math-ph↗
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↗
arXiv2019

Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees

Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular fulfills the spatial moment condition of Holmes and Perkins. Our method of proof is relatively simple, and, as we show, it applies also to oriented percolation and lattice trees. Via the convergence results of Holmes and Perkins, the upper bounds on the spatial moments can in fact be promoted to asymptotic formulas with explicit constants.

math.PR↗
arXiv2019

Critical two-point function for long-range models with power-law couplings: The marginal case for $d\ge d_c$

Consider the long-range models on $\mathbb{Z}^d$ of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as $|x|^{-d-α}$ for some $α>0$. In the previous work (Ann. Probab., 43, 639--681, 2015), we have shown in a unified fashion for all $α\ne2$ that, assuming a bound on the "derivative" of the $n$-step distribution (the compound-zeta distribution satisfies this assumed bound), the critical two-point function $G_{p_c}(x)$ decays as $|x|^{α\wedge2-d}$ above the upper-critical dimension $d_c\equiv(α\wedge2)m$, where $m=2$ for self-avoiding walk and the Ising model and $m=3$ for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the $n$-step distribution, that $G_{p_c}(x)$ for the marginal case $α=2$ decays as $|x|^{2-d}/\log|x|$ whenever $d\ge d_c$ (with a large spread-out parameter $L$). This solves the conjecture in the previous work, extended all the way down to $d=d_c$, and confirms a part of predictions in physics (Brezin, Parisi, Ricci-Tersenghi, J. Stat. Phys., 157, 855--868, 2014). The proof is based on the lace expansion and new convolution bounds on power functions with log corrections.

math-ph↗
arXiv2018

Application of the lace expansion to the $φ^4$ model

Using the Griffiths-Simon construction of the $φ^4$ model and the lace expansion for the Ising model, we prove that, if the strength $λ\ge0$ of nonlinearity is sufficiently small for a large class of short-range models in dimensions $d>4$, then the critical $φ^4$ two-point function $\langleφ_oφ_x\rangle_{μ_c}$ is asymptotically $|x|^{2-d}$ times a model-dependent constant, and the critical point is estimated as $μ_c=\mathscr{\hat J}-\frac\lambda2\langleφ_o^2\rangle_{μ_c}+O(λ^2)$, where $\mathscr{\hat J}$ is the massless point for the Gaussian model.

math-ph↗
arXiv2018

Hyperscaling for oriented percolation in 1+1 space-time dimensions

Consider nearest-neighbor oriented percolation in $d+1$ space-time dimensions. Let $ρ,η,ν$ be the critical exponents for the survival probability up to time $t$, the expected number of vertices at time $t$ connected from the space-time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality $dν\geη+2ρ$, which holds for all $d\ge1$ and is a strict inequality above the upper-critical dimension 4, becomes an equality for $d=1$, i.e., $ν=η+2ρ$, provided existence of at least two among $ρ,η,ν$. The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin, Tassion and Teixeira (2017).

math.PR↗
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↗
arXiv2017

Accurate estimation of interface phase diagram from STM images

We propose a new approach to constructing a phase diagram using the effective Hamiltonian derived only from a single real-space image produced by scanning tunneling microscopy (STM). Currently, there have been two main methods to construct phase diagrams in material science: ab initio calculations and CALPHAD with thermodynamic information obtained by experiments and/or theoretical calculations. Although the two methods have successfully revealed a number of unsettled phase diagrams, their results sometimes contradicted when it is difficult to construct an appropriate Hamiltonian that captures the characteristics of materials, e.g., for a system consisting of multiple-scale objects whose interactions are essential to the system's characteristics. Meanwhile, the advantage of our approach over existing methods is that it can directly and uniquely determine the effective Hamiltonian without any thermodynamic information. The validity of our approach is demonstrated through an Mg--Zn--Y long-period stacking-ordered structure, which is a challenging system for existing methods, leading to contradictory results. Our result successfully reproduces the ordering tendency seen in STM images that previous theoretical study failed to reproduce and clarifies its previously unknown phase diagram. Thus, our approach can be used to clear up contradictions shown by existing methods.

cond-mat.mtrl-sci↗
arXiv2016

The quenched critical point for self-avoiding walk on random conductors

Following similar analysis to that in Lacoin (PTRF 159, 777-808, 2014), we can show that the quenched critical point for self-avoiding walk on random conductors on the d-dimensional integer lattice is almost surely a constant, which does not depend on the location of the reference point. We provide its upper and lower bounds that are valid for all dimensions.

math.PR↗
arXiv2015

Critical two-point functions for long-range statistical-mechanical models in high dimensions

We consider long-range self-avoiding walk, percolation and the Ising model on $\mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)\asymp|x|^{-d-α}$ with $α>0$. The upper-critical dimension $d_{\mathrm{c}}$ is $2(α\wedge2)$ for self-avoiding walk and the Ising model, and $3(α\wedge2)$ for percolation. Let $α\ne2$ and assume certain heat-kernel bounds on the $n$-step distribution of the underlying random walk. We prove that, for $d>d_{\mathrm{c}}$ (and the spread-out parameter sufficiently large), the critical two-point function $G_{p_{\mathrm{c}}}(x)$ for each model is asymptotically $C|x|^{α\wedge2-d}$, where the constant $C\in(0,\infty)$ is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between $α<2$ and $α>2$. We also provide a class of random walks that satisfy those heat-kernel bounds.

math-ph↗
arXiv2011

Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation

We consider random walk and self-avoiding walk whose 1-step distribution is given by $D$, and oriented percolation whose bond-occupation probability is proportional to $D$. Suppose that $D(x)$ decays as $|x|^{-d-α}$ with $α>0$. For random walk in any dimension $d$ and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension $d_{\mathrm{c}}\equiv2(α\wedge2)$, we prove large-$t$ asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length $t$ or the average spatial size of an oriented percolation cluster at time $t$. This proves the conjecture for long-range self-avoiding walk in [Ann. Inst. H. Poincaré Probab. Statist. (2010), to appear] and for long-range oriented percolation in [Probab. Theory Related Fields 142 (2008) 151--188] and [Probab. Theory Related Fields 145 (2009) 435--458].

math.PR↗
arXiv2008

Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: I. The higher-point functions

We consider the critical spread-out contact process in Z^d with d\ge1, whose infection range is denoted by L\ge1. In this paper, we investigate the r-point function τ_{\vec t}^{(r)}(\vec x) for r\ge3, which is the probability that, for all i=1,...,r-1, the individual located at x_i\in Z^d is infected at time t_i by the individual at the origin o\in Z^d at time 0. Together with the results of the 2-point function in [van der Hofstad and Sakai, Electron. J. Probab. 9 (2004), 710-769; arXiv:math/0402049], on which our proofs crucially rely, we prove that the r-point functions converge to the moment measures of the canonical measure of super-Brownian motion above the upper-critical dimension 4. We also prove partial results for d\le4 in a local mean-field setting.

math.PR↗
arXiv2008

Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation

We prove that the Fourier transform of the properly-scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index α>0 converges to e^{-C|k|^{α\wedge2}} for some C\in(0,\infty) above the upper-critical dimension 2(α\wedge2). This answers the open question remained in the previous paper [arXiv:math/0703455]. Moreover, we show that the constant C exhibits crossover at α=2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients.

math.PR↗
arXiv2008

Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(α\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(α\wedge2)$ for percolation, where $d$ denotes the dimension and $α$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)

math.PR↗