KEIO RESEARCHERS INFORMATION SYSTEM

Profile 【 Display / hide 】

Profile 【 Display / hide 】

• My main research themes include the analysis of large deviation principles and probabilistic models such as first-passage percolation, directed polymers, the KPZ universality class, and the Ising perceptron.
  
  I received my Bachelor's degree from Nagoya University, and my Master's and Ph.D. degrees from the Research Institute for Mathematical Sciences at Kyoto University. Prior to my current position, I served as a postdoctoral researcher at the University of Basel in Switzerland and as a senior lecturer at Meiji University.

Message from the Faculty Member 【 Display / hide 】

Message from the Faculty Member 【 Display / hide 】

• Probability theory and statistical physics are fascinating fields that uncover universal and beautiful mathematical laws hidden within seemingly random and unpredictable phenomena.
  
  University mathematics, especially modern probability theory, requires more than just calculation skills. It demands the ability to rigorously formulate why such phenomena occur and to prove them logically. While you may find the material abstract and difficult at first, the joy of grasping the essential meaning of a theorem through trial and error is truly irreplaceable.
  
  Please do not hesitate to ask questions and actively engage in discussions whenever you encounter something you do not understand. Let us think together and enjoy exploring the depth of mathematics.

Career 【 Display / hide 】

Career 【 Display / hide 】

• 2020.04

  -

  2022.03

  University of Basel, Department of Mathematics and Computer Science, Postdoctoral researcher

• 2022.04

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  2025.08

  Meiji University, Department of Mathematics, School of Science and Technology, Senior Lecturer

Academic Background 【 Display / hide 】

Academic Background 【 Display / hide 】

• 2016.04

  -

  2019.03

  Kyoto University, Department of science, Mathematics

  Graduate School, Completed, Doctoral course

Academic Degrees 【 Display / hide 】

Academic Degrees 【 Display / hide 】

• Doctor of Science, Kyoto University, Coursework, 2019.03

  Maximal edge-traversal time in First Passage Percolation

Research Areas 【 Display / hide 】

Research Areas 【 Display / hide 】

• Natural Science / Applied mathematics and statistics (Probability theory)

Papers 【 Display / hide 】

Papers 【 Display / hide 】

• 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

  ISSN  09446532

  　View Summary

  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

  ISSN  01788051

  　View Summary

  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 2025.04

  ISSN  10635203

  　View Summary

  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 )  2025.03

  ISSN  13850172

  　View Summary

  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

  ISSN  26900998

  　View Summary

  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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Awards 【 Display / hide 】

Awards 【 Display / hide 】

• The MSJ Takebe Katahiro Prize for Encouragement of Young Researchers

  Shuta Nakajima, 2018.10, 一般社団法人 日本数学会, 最速浸透問題の研究

  Type of Award： International academic award (Japan or overseas)

Courses Taught 【 Display / hide 】

Courses Taught 【 Display / hide 】

• MATHEMATICAL SCIENCES PRACTICAL RESEARCH ACTIVITY A

  2026

• DOCTORAL RESEARCH ON MATHEMATICAL AND PHYSICAL SCIENCES

  2026

• GRADUATE RESEARCH ON FUNDAMENTAL SCIENCE AND TECHNOLOGY 1

  2026

• PROBABILITY THEORY 1 AND ITS EXERCISE

  2026

• MATHEMATICAL SCIENCES PRACTICAL RESEARCH ACTIVITY B

  2026

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