SAKAI Akira Professor

Applied Mathematics

Department of Mathematics
Research Interest
Probability theory, Statistical mechanics
Critical phenomena, Interacting particle systems, Ising-Potts model, Lace expansion, Percolation, Phase transitions

Research Activities

My major research field is mathematical physics (probability and statistical mechanics). The topics I have been most fascinated with are phase transitions and critical phenomena, as well as associated scaling limits. For example, the Ising model, a statistical-mechanical model of ferromagnetism, is known to take on positive spontaneous magnetization as soon as the temperature of the system is turned down below the critical point. Various other observables also exhibit singular behavior around the critical point, due to cooperation of infinitely many interacting variables. To fully understand such phenomena, it would require development of a theory beyond the standard probability theory. This is a challenging and intriguing problem, towards which I would love to make even a tiny contribution.

The mathematical models I have been studying are

  • the Ising model,
  • the φ4 model (in lattice scalar-field theory),
  • self-avoiding walk (a model for linear polymers),
  • percolation (for random media),
  • the contact process (for the spread of an infectious disease),
  • random walk with reinforcement,
  • stochastic cellular automata.


  • L.-C. Chen and A. Sakai,
    Critical two-point function for long-range models with power-law couplings: the marginal case for d ≥ dc,
    Comm. Math. Phys. 372 (2019): 543-572.
  • A. Sakai,
    Application of the lace expansion to the φ4 model,
    Comm. Math. Phys. 336 (2015): 619-648.
  • L.-C. Chen and A. Sakai,
    Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation,
    Ann. Probab. 39 (2011) 507-548
  • A. Sakai,
    Lace expansion for the Ising model,
    Comm. Math. Phys. 272 (2007): 283-344.
  • R. van der Hofstad and A. Sakai,
    Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions,
    Probab. Theory Relat. Fields 132 (2005): 438-470.