arXiv subjects

Shuta Nakajima

Publications and source records attributed to Shuta Nakajima.

At least 19 recordsLinked to original sources

arXiv2026

A quantitative replica-symmetric bound for Sherrington--Kirkpatrick model in the entire de Almeida--Thouless region

We consider the Sherrington--Kirkpatrick model with inverse temperature $\beta>0$ and deterministic external field $h>0$. Let $q$ be the replica-symmetric fixed point: $q=\mathbb E{\rm tanh}^2 (h+\beta\sqrt q\,Z)$, where $Z$ is a standard normal. We prove that, uniformly on compact subsets of the strict de Almeida--Thouless region: $\beta^2\mathbb E{\rm sech}^4(h+\beta\sqrt{q} Z) <1,$ the overlap satisfies the concentration: $$ \mathbb E\langle (R_{12}-q)^2\rangle=O(N^{-1}). $$ As a consequence, we obtain an $O(N^{-1})$ replica-symmetric free-energy correction and identify the finite-volume replicon susceptibility. Our proof is self-contained and does not use the identification of the limiting free energy with the Parisi variational formula. Moreover, we prove the central limit theorem for the overlap in this region. The present paper provides an alternative proof of the replica-symmetric free energy formula in the de Almeida--Thouless region, which was recently established by Lopatto [arXiv:2604.11921]. An advantage of our approach is that it establishes an explicit quantitative bound and yields the replica-symmetric free energy formula as a consequence. The main result supersedes the corresponding result in our recent preprint arXiv:2607.23427, extending the replica-symmetric bounds to the strict de Almeida-Thouless region. However, we keep the previous preprint, since its argument is different and substantially simpler than the one given here.

math.PR↗

arXiv2026

On the most likely geodesic in last passage percolation

We consider the problem of identifying the paths most likely to occur as geodesics in last passage percolation. Heuristics suggest that these modal paths should be the extreme corner paths, going straight between the corners of the cube of accessible vertices. We identify three mechanisms which favour such corner paths, and show that they are more likely to appear than all but a vanishingly small portion of paths. We make more specific comparisons in exponential last passage percolation, where moderate deviation estimates may be used to show that corner paths are nearly modal in a precise sense. Finally, we show a form of monotonicity of geodesic probabilities in a special case and conjecture that this holds in general.

math.PR↗

arXiv2026

A note on Lata\la's argument in SK model

In this note, we consider the Sherrington--Kirkpatrick model with deterministic external field. Let $q=q(\beta,h)$ denote the solution of the replica-symmetric self-consistency equation \[ q=\mathbb E\tanh^2\!\left(h+\beta\sqrt q\,Z\right), \qquad Z\sim N(0,1), \] where $\beta$ and $h$ are inverse temperature and external field, respectively. By refining Lata\la' s argument, previously limited to $\beta < \frac{1}{2}$, and using the Kearns--Saul inequality, we prove overlap concentration and convergence of the free energy to the replica symmetric formula with error $O(N^{-1})$ whenever \[ \beta^2\frac{q}{{\rm arctanh}q}<1. \] Note that for any $\beta<1$ and $h\in \mathbb R$, the condition above is satisfied. Moreover, for every nonzero $h$, this region contains a nonempty interval with $\beta>1$.

math.PR↗

arXiv2026

Sharp behavior of the free energy for the two-dimensional directed polymer model

We consider the directed polymer model on $\mathbb{Z}^d$, in an i.i.d.\ random environment $\omega=(\omega_{n,x})_{n\geq 0,x\in\mathbb Z^d}$, focusing on the critical dimension $d=2$. Our main contribution is to give a sharp lower bound on the free energy in the high-temperature regime. Our proof uses a percolation argument inspired by Lacoin (2010), for which we introduce a key property of bounded ``$\log$-energy'': this property quantifies the regularity of the polymer measures at diffusive scales and we show that it propagates along open paths. Writing $\mathfrak{f}(\beta)$ for the quenched free energy, and setting $\lambda(\beta):=\log \mathbb E[e^{\beta\omega_{1,0}}]$ and $\sigma(\beta)^2:=e^{\lambda(2\beta)-2\lambda(\beta)}-1$, our lower bound combined with Theorem 2.8 of Berger, Caravenna, and Turchi (2025) gives $$ -\mathfrak{f}(\beta) \asymp \exp{\Big(- \frac{\pi}{\sigma^2(\beta)}\Big)},\quad \text{ as $\beta\downarrow 0$.} $$

math.PR↗

arXiv2025

Uniqueness of Replica-symmetric Saddle Point for Ising Perceptron

We study the replica-symmetric saddle point equations for the Ising perceptron with Gaussian disorder and margin $κ\ge 0$. We prove that for each $κ\ge 0$ there is a critical capacity $α_c(κ)=\frac{2}{π\,\mathbb E[(κ-Z)_+^2]}$, where $Z$ is a standard normal and $(x)_+=\max\{x,0\}$, such that the saddle point equation has a unique solution for $α\in(0,α_c(κ))$ and has no solution when $α\ge α_c(κ)$. When $α\uparrow α_c(κ)$ and $κ>0$, the replica-symmetric free energy at this solution diverges to $-\infty$. In the zero-margin case $κ=0$, Ding and Sun obtained a conditional uniqueness result, with one step verified numerically. Our argument gives a fully analytic proof without computer assistance. We used GPT-5 to help develop intermediate proof steps and to perform sanity-check computations.

math.PR↗

arXiv2025

Waveguide-array-based multiplexed photonic interface for atom array

The growing demand for high-capacity quantum communication and large-scale quantum computing underscores the importance of networking quantum processing units via multiplexed photonic channels. A neutral atom array with multiplexed atom-photon entanglement is a promising platform for its realization. Here, we demonstrate a key multiplexed photonic interface guiding the photons from an atom array to a single-mode waveguide array fabricated on a glass-based photonic integrated circuit. Remarkable 10 channels out of the 32-channel waveguide array with 25 $\mu$m pitch couple to photons from 10 sites of the atom array with Rydberg gate-enabled separation. Based on the observed correlation between the atomic states and the polarization of the photon with a visibility of 0.87, we anticipate its applicability to a large-scale multiplexed atom-photon entanglement generation for networking quantum processing units.

quant-ph↗

arXiv2025

Moderate deviations in first-passage percolation for bounded weights

We investigate the moderate and large deviations in first-passage percolation (FPP) with bounded weights on $\mathbb{Z}^d$ for $d \geq 2$. Write $T(\mathbf{x}, \mathbf{y})$ for the first-passage time and denote by $μ(\mathbf{u})$ the time constant in direction $\mathbf{u}$. In this paper, we establish that, if one assumes that the sublinear error term $T(\mathbf{0}, N\mathbf{u}) - Nμ(\mathbf{u})$ is of order $N^χ$, then under some unverified (but widely believed) assumptions, for $χ< a < 1$, \begin{align*} &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) > Nμ(\mathbf{u}) + N^a\bigr) = \exp{\Big(-\,N^{\frac{d(1+o(1))}{1-χ}(a-χ)}\Big)},\end{align*} \begin{align*} &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) < Nμ(\mathbf{u}) - N^a\bigr) = \exp{\Big(-\,N^{\frac{1+o(1)}{1-χ}(a-χ)}\Big)}, \end{align*} with accompanying estimates in the borderline case $a=1$. Moreover, the exponents $\frac{d}{1-χ}$ and $\frac{1}{1-χ}$ also appear in the asymptotic behavior near $0$ of the rate functions for upper and lower tail large deviations. Notably, some of our estimates are established rigorously without relying on any unverified assumptions. Our main results highlight the interplay between fluctuations and the decay rates of large deviations, and bridge the gap between these two regimes. A key ingredient of our proof is an improved concentration via multi-scale analysis for several moderate deviation estimates, a phenomenon that has previously appeared in the contexts of two-dimensional last-passage percolation and two-dimensional rotationally invariant FPP.

math.PR↗

arXiv2025

Quantum Chaos, Thermalization, and Non-locality

In this paper, we numerically investigate whether quantum thermalization occurs during the time evolution induced by a non-local Hamiltonian whose spectra exhibit integrability. This non-local and integrable Hamiltonian is constructed by combining two types of integrable Hamiltonians. From the time dependence of entanglement entropy and mutual information, we find that non-locality can evolve the system into the typical state. On the other hand, the time dependence of logarithmic negativity shows that the non-locality can destroy the quantum correlation. These findings suggest that the quantum thermalization induced by the non-local Hamiltonian does not require the quantum chaoticity of the system.

hep-th↗

arXiv2025

Equivalence of fluctuations of discretized SHE and KPZ equations in the subcritical weak disorder regime

We study the fluctuations of discretized versions of the stochastic heat equation (SHE) and the Kardar-Parisi-Zhang (KPZ) equation in spatial dimensions $d\geq 3$ in the weak disorder regime. The discretization is defined using the directed polymer model. Previous research has identified the scaling limit of both equations under a suboptimal moment condition and, in particular, it was established that both converge in law to the same limit. We extend this result by showing that the fluctuations of both equations are close in probability in the subcritical weak disorder regime, indicating that they share the same scaling limit (the existence of which remains open). Our result applies under a moment condition that is expected to hold throughout the interior of the weak disorder phase, which is currently only known under a technical assumption on the environment. We also prove a lower tail concentration of the partition functions.

math.PR↗

arXiv2025

High moments of 2d directed polymers up to quasi-criticality

We consider two-dimensional directed polymers in random environment in the sub-critical regime and in the quasi-critical regime introduced recently by Caravenna, Cottini and Rossi, arXiv:2307.02453v1. For $q\leq q_N$ with $q_N\to\infty$ diverging at a suitable rate with the size of the system, we obtain upper bound estimates on the $q$-moment of the partition function for general environments. In the sub-critical regime, our results improve the $q_N$-threshold obtained for Gaussian environment in Cosco, Zeitouni, Comm. Math. Phys (2023). As a corollary, we derive large deviation estimates with a Gaussian rate function.

math.PR↗

arXiv2025

The maximum of the two dimensional Gaussian directed polymer in the subcritical regime

We study the maximum $ϕ_N^*$ of the partition function of the two dimensional (subcritical) Gaussian directed polymer over an $\sqrt N \times \sqrt N$ box. We show that $ϕ_N^*/\log N$ converges towards a constant $σ^*$, which we identify to be the same as for the maximum of a branching random walk with a slowly varying variance profile as studied in Fang-Zeitouni, J. Stat. Phys. 2012 and (in the context of the generalized random energy model) in Bovier-Kurkova, Ann. Inst. H. Poincare 2004.

math.PR↗

arXiv2024

Injectivity of ReLU networks: perspectives from statistical physics

When can the input of a ReLU neural network be inferred from its output? In other words, when is the network injective? We consider a single layer, $x \mapsto \mathrm{ReLU}(Wx)$, with a random Gaussian $m \times n$ matrix $W$, in a high-dimensional setting where $n, m \to \infty$. Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for $α= \frac{m}{n}$ by studying the expected Euler characteristic of a certain random set. We adopt a different perspective and show that injectivity is equivalent to a property of the ground state of the spherical perceptron, an important spin glass model in statistical physics. By leveraging the (non-rigorous) replica symmetry-breaking theory, we derive analytical equations for the threshold whose solution is at odds with that from the Euler characteristic. Furthermore, we use Gordon's min--max theorem to prove that a replica-symmetric upper bound refutes the Euler characteristic prediction. Along the way we aim to give a tutorial-style introduction to key ideas from statistical physics in an effort to make the exposition accessible to a broad audience. Our analysis establishes a connection between spin glasses and integral geometry but leaves open the problem of explaining the discrepancies.

cond-mat.dis-nn↗

arXiv2024

Lipschitz-continuity of time constant in generalized First-passage percolation

In this article, we consider a generalized First-passage percolation model, where each edge in $\mathbb{Z}^d$ is independently assigned an infinite weight with probability $1-p$, and a random finite weight otherwise. The existence and positivity of the time constant have been established in [CT16]. Recently, using sophisticated multi-scale renormalizations, Cerf and Dembin [CD22] proved that the time constant of chemical distance in super-critical percolation is Lipschitz continuous. In this work, we propose a different approach leveraging lattice animal theory and a simple one-step renormalization with the aid of Russo's formula, to show the Lipschitz continuity of the time constant in generalized First-passage percolation.

math.PR↗

arXiv2024

Lipschitz-type estimate for the frog model with Bernoulli initial configuration

We consider the frog model with Bernoulli initial configuration, which is an interacting particle system on the multidimensional lattice consisting of two states of particles: active and sleeping. Active particles perform independent simple random walks. On the other hand, although sleeping particles do not move at first, they become active and can move around when touched by active particles. Initially, only the origin has one active particle, and the other sites have sleeping particles according to a Bernoulli distribution. Then, starting from the original active particle, active ones are gradually generated and propagate across the lattice, with time. It is of interest to know how the propagation of active particles behaves as the parameter of the Bernoulli distribution varies. In this paper, we treat the so-called time constant describing the speed of propagation, and prove that the absolute difference between the time constants for parameters $p,q \in (0,1]$ is bounded from above and below by multiples of $|p-q|$.

math.PR↗

arXiv2023

Upper tail large deviation for the one-dimensional frog model

In this paper, we study the upper tail large deviation for the one-dimensional frog model. In this model, sleeping and active frogs are assigned to vertices on $\mathbb Z$. While sleeping frogs do not move, the active ones move as independent simple random walks and activate any sleeping frogs. The main object of interest in this model is the asymptotic behavior of the first passage time ${\rm T}(0,n)$, which is the time needed to activate the frog at the vertex $n$, assuming there is only one active frog at $0$ at the beginning. While the law of large numbers and central limit theorems have been well established, the intricacies of large deviations remain elusive. Using renewal theory, Bérard and Ramírez have pointed out a slowdown phenomenon where the probability that the first passage time ${\rm T}(0,n)$ is significantly larger than its expectation decays sub-exponentially and lies between $\exp(-n^{1/2+o(1)})$ and $\exp(-n^{1/3+o(1)})$. In this article, using a novel covering process approach, we confirm that $1/2$ is the correct exponent, i.e., the rate of upper large deviations is given by $n^{1/2}$. Moreover, we obtain an explicit rate function that is characterized by properties of Brownian motion and is strictly concave.

math.PR↗

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↗