慶應義塾研究者情報データベース

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• 最速浸透問題、有向ポリマー、KPZ普遍クラス、イジングパーセプトロンなどの確率モデルや大偏差原理の解析を主な研究テーマとしています。
  
  名古屋大学で学士号、京都大学数理解析研究所で修士号および博士号を取得し、バーゼル大学（スイス）でのポスドク研究員や明治大学での専任講師などを経て現職に至ります。

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• 確率論や統計物理は、一見すると不規則で予測不可能な現象の中に、普遍的で美しい数学的法則を見出していく非常に魅力的な分野です。
  
  大学での数学、特に現代の確率論では、単なる計算の技術だけではなく、「なぜそのような現象が起きるのか」を厳密に定式化し、論理的に証明していく力が求められます。最初は抽象的で難しく感じるかもしれませんが、試行錯誤を重ねて定理の本質的な意味を理解できたときの喜びは、何物にも代えがたいものです。
  
  分からないことがあれば、ぜひ積極的に質問し、議論してください。共に考え、数学の広がりと奥深さを楽しみながら学んでいきましょう。

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• 2020年04月

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  2022年03月

  バーゼル大学, 数理情報学部, ポスドク研究員

• 2022年04月

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  2025年08月

  明治大学, 理工学部　数学科, 専任講師

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• 2016年04月

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  2019年03月

  京都大学, 理学部, 数学

  大学院, 修了, 博士

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• 博士（理学）, 京都大学, 課程, 2019年03月

  ファーストパッセージパーコレーションの最大辺移動時間

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• 自然科学一般 / 応用数学、統計数学 （確率論）

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• Asymptotics of the p-Capacity in the Critical Regime

  Cosco C., Nakajima S., Schweiger F.

  Journal of Convex Analysis 33 （ 1-2 ） 13 - 27 2026年

  ISSN  09446532

  　概要を見る

  We are interested in the asymptotics of the p-capacity between the origin and the set nB, where B is the boundary of the unit ball of the lattice Z^d. The p-capacity is defined as the minimum of the Dirichlet energy associated with a discrete version of the p-Laplacian. This variational problem has arisen in particular in the study of large deviations for first passage percolation. For p < d, the p-capacity converges to some positive constant, while 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<inf>d</inf>(log n)^−d+1 with an explicit constant c<inf>d</inf>. Our proof relies on Thomson’s principle for the p-capacity.

• Upper tail large deviation for the one-dimensional frog model

  Can V.H., Kubota N., Nakajima S.

  Probability Theory and Related Fields 2026年

  ISSN  01788051

  　概要を見る

  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 sites of Z. While sleeping frogs do not move, the active ones move as independent simple random walks and activate any sleeping frogs. An interest of this model is the asymptotic behavior of the first passage time T(0,n), which is the time needed to activate the sleeping frog at the site 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, J., Ramírez, A.F. (Ann. Probab. 44(4), 2770–2816, 2016) have pointed out a slowdown phenomenon where the probability that the first passage time T(0,n) is significantly larger than its expectation decays sub-exponentially and lies between exp(-n1/2+o(1)) and exp(-n1/3+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 n1/2. Moreover, we obtain an explicit rate function that is characterized by properties of Brownian motion and is strictly concave.

  • Access to Document (DOI)

• Injectivity of ReLU networks: Perspectives from statistical physics

  Maillard A., Bandeira A.S., Belius D., Dokmanić I., Nakajima S.

  Applied and Computational Harmonic Analysis 76 2025年04月

  ISSN  10635203

  　概要を見る

  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↦ReLU(Wx), with a random Gaussian m×n matrix W, in a high-dimensional setting where n,m→∞. Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for α=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.

  • Access to Document (DOI)

• Lipschitz-Type Estimate for the Frog Model with Bernoulli Initial Configuration

  Can V.H., Kubota N., Nakajima S.

  Mathematical Physics Analysis and Geometry 28 （ 1 ）  2025年03月

  ISSN  13850172

  　概要を見る

  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∈(0,1] is bounded from above and below by multiples of |p-q|.

  • Access to Document (DOI)

• EQUIVALENCE OF FLUCTUATIONS OF DISCRETIZED SHE AND KPZ EQUATIONS IN THE SUBCRITICAL WEAK DISORDER REGIME

  Junk S., Nakajima S.

  Probability and Mathematical Physics 6 （ 3 ） 819 - 856 2025年

  ISSN  26900998

  　概要を見る

  We study the fluctuations of discretized versions of the stochastic heat equation (SHE) and the Kardar– Parisi–Zhang (KPZ) equation in spatial dimensions d ≥ 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.

  • Access to Document (DOI)

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• 2018年度日本数学会賞建部賢弘奨励賞

  中島秀太, 2018年10月, 一般社団法人 日本数学会, 最速浸透問題の研究

  受賞区分： 国内外の国際的学術賞

担当授業科目 【 表示 ／ 非表示 】

担当授業科目 【 表示 ／ 非表示 】

• 数理科学実践研究活動Ａ

  2026年度

• 先端数物科学博士研究

  2026年度

• 基礎理工学特別研究第１

  2026年度

• 確率論第１同演習

  2026年度

• 数理科学実践研究活動Ｂ

  2026年度

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