arXiv subjects

Shuta Nakajima

Publications and source records attributed to Shuta Nakajima.

At least 37 records · Page 2Linked to original sources

arXiv2023

Designing nontrivial one-dimensional Floquet topological phases using a spin-1/2 double-kicked rotor

A quantum kicked rotor model is one of the promising systems to realize various Floquet topological phases. We consider a double-kicked rotor model for a one-dimensional quasi-spin-1/2 Bose-Einstein condensate with spin-dependent and spin-independent kicks which are implementable for cold atomic experiments. We theoretically show that the model can realize all the Altland-Zirnbauer classes with nontrivial topology in one dimension. In the case of class CII, we show that a pair of winding numbers $(w_0,w_π)\in 2\mathbb{Z}\times 2\mathbb{Z}$ featuring the edge states at zero and $π$ quasienergy, respectively, takes various values depending on the strengths of the kicks. We also find that the winding numbers change to $\mathbb{Z}$ when we break the time-reversal and particle-hole symmetries by changing the phase of a kicking lattice. We numerically confirm that the winding numbers can be obtained by measuring the mean chiral displacement in the long-time limit in the present case with four internal degrees of freedom. We further propose two feasible methods to experimentally realize the spin-dependent and spin-independent kicks required for various topological phases.

cond-mat.quant-gas↗

arXiv2023

Asymptotics of the $p$-capacity in the critical regime

In this note, we are interested in the asymptotics as $n\to\infty$ of the $p$-capacity between the origin and the set $nB$, where $B$ is the boundary of the unit ball of the lattice $\mathbb Z^d$. The $p$-capacity is defined as the minimum of the Dirichlet energy $\frac{1}{2}\sum_{x\in \mathbb Z^d} \sum_{y\sim x} |f(x)-f(y)|^{p}$ with $f$ subject to the boundary conditions $f(0)=0$ and $f\geq 1$ on $nB$. This variational problem has arisen in particular in the study of large deviations for first passage percolation. For $p d$ the capacity vanishes polynomially fast. The present paper deals with the case $p=d$, for which we prove that the $p$-capacity vanishes as $c_d (\log n)^{-d+1}$ with an explicit constant $c_d$.

math.AP↗

arXiv2023

On the upper tail large deviation rate function for chemical distance in supercritical percolation

We consider the supercritical bond percolation on $\mathbb Z^d$ and study the graph distance on the percolation graph called the chemical distance. It is well-known that there exists a deterministic constant $μ(x)$ such that the chemical distance $\mathcal D(0,nx)$ between two connected points $0$ and $nx$ grows like $nμ(x)$. Garet and Marchand (Ann. Prob., 2007) proved that the probability of the upper tail large deviation event $\left\{nμ(x)(1+\varepsilon)<\mathcal D(0,nx)<\infty\right\} $ decays exponentially with respect to $n$. In this paper, we prove the existence of the rate function for upper tail large deviation when $d\ge 3$ and $\varepsilon>0$ is small enough. Moreover, we show that for any $\varepsilon>0$, the upper tail large deviation event is created by space-time cut-points (points that all paths from $0$ to $nx$ must cross after a given time) that force the geodesics to consume more time by going in a non-optimal direction or by wiggling considerably. This enables us to express the rate function in regards to space-time cut-points.

math.PR↗

arXiv2023

A variational formula for large deviations in First-passage percolation under tail estimates

Consider first passage percolation with identical and independent weight distributions and first passage time ${\rm T}$. In this paper, we study the upper tail large deviations $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$, for $ξ>0$ and $x\neq 0$ with a time constant $μ$ and a dimension $d$, for weights that satisfy a tail assumption $ β_1\exp{(-αt^r)}\leq \mathbb P(τ_e>t)\leq β_2\exp{(-αt^r)}.$ When $r\leq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $\exp{(-(2dξ+o(1))n)}$. When $1< r\leq d$, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. For $r n(μ+ξ)$ is described by a localization of high weights around the origin. The picture changes for $r\geq d$ where the configuration is not anymore localized.

math.PR↗

arXiv2022

Sharp threshold sequence and universality for Ising perceptron models

We study a family of Ising perceptron models with $\{0,1\}$-valued activation functions. This includes the classical half-space models, as well as some of the symmetric models considered in recent works. For each of these models we show that the free energy is self-averaging, there is a sharp threshold sequence, and the free energy is universal with respect to the disorder. A prior work of Xu (2019) used very different methods to show a sharp threshold sequence in the half-space Ising perceptron with Bernoulli disorder. Recent works of Perkins--Xu (2021) and Abbe--Li--Sly (2021) determined the sharp threshold and limiting free energy in a symmetric perceptron model. The results of this paper apply in more general settings, and are based on new "add one constraint" estimates extending Talagrand's estimates for the half-space model (1999, 2011).

math.PR↗

arXiv2021

Gardner formula for Ising perceptron models at small densities

We consider the Ising perceptron model with N spins and M = N*alpha patterns, with a general activation function U that is bounded above. For U bounded away from zero, or U a one-sided threshold function, it was shown by Talagrand (2000, 2011) that for small densities alpha, the free energy of the model converges in the large-N limit to the replica symmetric formula conjectured in the physics literature (Krauth--Mezard 1989, see also Gardner--Derrida 1988). We give a new proof of this result, which covers the more general class of all functions U that are bounded above and satisfy a certain variance bound. The proof uses the (first and second) moment method conditional on the approximate message passing iterates of the model. In order to deduce our main theorem, we also prove a new concentration result for the perceptron model in the case where U is not bounded away from zero.

math.PR↗

arXiv2021

Competition and interplay between topology and quasi-periodic disorder in Thouless pumping of ultracold atoms

Robustness against perturbations lies at the heart of topological phenomena. If, however, a perturbation such as disorder becomes dominant, it may cause a topological phase transition between topologically non-trivial and trivial phases. Here we experimentally reveal the competition and interplay between topology and quasi-periodic disorder in a Thouless pump realized with ultracold atoms in an optical lattice, by creating a quasi-periodic potential from weak to strong regimes in a controllable manner. We demonstrate a disorder-induced pumping in which the presence of quasi-periodic disorder can induce a non-trivial pump for a specific pumping sequence, while no pump is observed in the clean limit. Our highly controllable system, which can also straightforwardly incorporate interatomic interaction, could be a unique platform for studying various disorder-related novel effects in a wide range of topological quantum phenomena.

cond-mat.quant-gas↗

arXiv2021

Triviality of the geometry of mixed $p$-spin spherical Hamiltonians with external field

We study isotropic Gaussian random fields on the high-dimensional sphere with an added deterministic linear term, also known as mixed p-spin Hamiltonians with external field. We prove that if the external field is sufficiently strong, then the resulting function has trivial geometry, that is only two critical points. This contrasts with the situation of no or weak external field where these functions typically have an exponential number of critical points. We give an explicit threshold $h_c$ for the magnitude of the external fieldnecessary for trivialization and conjecture $h_c$ to be sharp. The Kac-Rice formula is our main tool. Our work extends [Fyo15], which identified the trivial regime for the special case of pure p-spin Hamiltonians with random external field.

math.PR↗

arXiv2021

Fluctuations of two-dimensional stochastic heat equation and KPZ equation in subcritical regime for general initial conditions

The solution of Kardar-Parisi-Zhang equation (KPZ equation) is solved formally via Cole-Hopf transformation $h=\log u$, where $u$ is the solution of multiplicative stochastic heat equation(SHE). In earlier works by Chatterjee and Dunlap, Caravenna, Sun, and Zygouras, and Gu, they consider the solution of two dimensional KPZ equation via the solution $u_\varepsilon$ of SHE with flat initial condition and with noise which is mollified in space on scale in $\varepsilon$ and its strength is weakened as $β_\varepsilon=\hatβ \sqrt{\frac{2π\varepsilon}{-\log \varepsilon}}$, and they prove that when $\hatβ\in (0,1)$, $\frac{1}{β_\varepsilon}(\log u_\varepsilon-\mathbb{E}[\log u_\varepsilon])$ converges in distribution to a solution of Edward-Wilkinson model as a random field. In this paper, we consider a stochastic heat equation $u_\varepsilon$ with general initial condition $u_0$ and its transformation $F(u_\varepsilon)$ for $F$ in a class of functions $\mathfrak{F}$, which contains $F(x)=x^p$ ($0<p\leq 1$) and $F(x)=\log x$. Then, we prove that $\frac{1}{β_\varepsilon}(F(u_\varepsilon(t,x))-\mathbb{E}[F(u_\varepsilon(t,x))])$ converges in distribution to Gaussian random variables jointly in finitely many $F\in \mathfrak{F}$, $t$, and $u_0$. In particular, we obtain the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depends on $u_0$. Our main tools are Itô's formula, the martingale central limit theorem, and the homogenization argument as in the works by Cosco and the authors. To this end, we also prove the local limit theorem for the partition function of intermediate $2d$-directed polymers

math.PR↗

arXiv2021

Limiting results for the free energy of directed polymers in random environment with unbounded jumps

We study asymptotics of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by Bernoulli variables. We first establish the existence and continuity of the free energy including the negative infinity value of the coupling constant $β$. Our proof of existence at $β=-\infty$ differs from existing ones in that it avoids the direct use of subadditivity. Secondly, we identify the asymptotics of the free energy at $β=-\infty$ in the limit of the success probability of the Bernoulli variables tending to one. It is described by using the so-called time constant of a certain directed first passage percolation. Our proof relies on a certain continuity property of the time constant, which is of independent interest.

math.PR↗

arXiv2021

Maximal edge-traversal time in First Passage Percolation

In this paper, we study the maximal edge-traversal time (simply we call maximal weight hereafter) on the optimal paths in the first passage percolation for several edge distributions, including the Pareto and Weibull distributions. It is known to be unbounded when the edge distribution has unbounded support [J. van den Berg and H. Kesten. Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 56-80, 1993]. We determine the order of the growth up to a multiplicative constant.

math.PR↗

arXiv2020

Divergence of non-random fluctuation for Euclidean first-passage percolation

The non-random fluctuation is one of the central objects in first passage percolation. It was proved in [Shuta Nakajima. Divergence of non-random fluctuation in First Passage Percolation. {\em Electron. Commun. Probab.} 24 (65), 1-13. 2019.] that for a particular asymptotic direction, it diverges in a lattice first passage percolation with an explicit lower bound. In this paper, we discuss the non-random fluctuation in Euclidean first passage percolations and show that it diverges in dimension $d\geq 2$ in this model also. Compared with the result in \cite{N19}, the present result is proved for any direction and improves the lower bound.

math.PR↗

arXiv2020

Topological Thouless Pumping of Ultracold Fermions

A gas of electrons in a one-dimensional periodic potential can be transported even in the absence of a voltage bias if the potential is modulated slowly and periodically in time. Remarkably, the transferred charge per cycle is only sensitive to the topology of the path in parameter space. Although this so-called Thouless charge pump has first been proposed more than thirty years ago, it has not yet been realized. Here we report the first demonstration of topological Thouless pumping using ultracold atoms in a dynamically controlled optical superlattice. We observe a shift of the atomic cloud as a result of pumping and extract the topological invariance of the pumping process from this shift. We demonstrate the topological nature of the Thouless pump by varying the topology of the pumping path and verify that the topological pump indeed works in the quantum region by varying speed and temperature.

cond-mat.quant-gas↗

arXiv2020

Gaussian fluctuations for the directed polymer partition function for $d\geq 3$ and in the whole $L^2$-region

We consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations of the tail $n^{(d-2)/4}(W_\infty - W_n)$ of the normalized partition function. It was proven by Comets and Liu, that for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting partition function and an independent Gaussian random variable. We extend the result to the whole $L^2$-region, which is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the considered scaling. To do so, we manage to avoid the heavy 4th-moment computation and instead rely on the local limit theorem for polymers and homogenization.

math.PR↗

arXiv2020

Law of large numbers and fluctuations in the sub-critical and $L^2$ regions for SHE and KPZ equation in dimension $d\geq 3$

There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar-Parisi-Zhang (KPZ) equation in dimension $d\geq 3$ as the smoothing parameter is switched off, but most of the results did not hold in the full temperature regions where they should. Inspired by martingale techniques coming from the directed polymers literature, we first extend the law of large numbers for SHE obtained in [MSZ16] to the full weak disorder region of the associated polymer model and to more general initial conditions. We further extend the Edwards-Wilkinson regime of the SHE and KPZ equation studied in [GRZ18,MU17,DGRZ20] to the full $L^2$-region, along with multidimensional convergence and general initial conditions for the KPZ equation (and SHE), which were not proven before. To do so, we rely on a martingale CLT combined with a refinement of the local limit theorem for polymers.

math.PR↗

arXiv2019

Upper tail large deviations for a class of distributions in First-passage percolation

In this paper we consider the first passage percolation with identical and independent exponentially distributions, called the Eden growth model, and we study the upper tail large deviations for the first passage time ${\rm T}$. Our main results prove that for any $ξ>0$ and $x\neq 0$, $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$ decays as $\exp{(-(2dξ+o(1))n)}$ with a time constant $μ$ and a dimension $d$. Moreover, we extend the result to stretched exponential distributions. On the contrary, we construct a continuous distribution with a finite exponential moment where the rate function does not exist.

math.PR↗

arXiv2019

Divergence of shape fluctuation for general distributions in first passage percolation

We study the shape fluctuation in the first passage percolation on $\mathbb{Z}^d$. It is known that it diverges when the distribution obeys Bernoulli in [Yu Zhang. The divergence of fluctuations for shape in first passage percolation. Probab. Theory. Related. Fields. 136(2) 298-320, 2006]. In this paper, we extend the result to general distributions.

math.PR↗

arXiv2019

Divergence of non-random fluctuation in First Passage Percolation

We study non-random fluctuation in the first passage percolation on $\mathbb{Z}^d$ and show that it diverges for any dimension. We also prove the divergence of the non-random shape fluctuation, which was conjectured in [Yu Zhang. The divergence of fluctuations for shape in first passage percolation. {\em Probab. Theory. Related. Fields.} 136(2) 298--320, 2006].

math.PR↗