SAKAI, Akira

SAKAI, Akira
Research Field
Applied Mathematics
Position
Associate Professor
Organization
Department of Mathematics
Research Interest
Probability theory, Statistical mechanics
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.

Papers:
[1] Y. Chino and A. Sakai.
The quenched critical point for self-avoiding walk on random conductors.
J. Stat. Phys. 163 (2016): 754-764.

[2] A. Sakai.
Application of the lace expansion to the φ4 model.
Commun. Math. Phys. 336 (2015): 619-648.

[3] L.-C. Chen and A. Sakai.
Critical two-point functions for long-range statistical-mechanical models in high dimensions.
Ann. Probab. 43 (2015): 639-681.

[4] 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

[5] A. Sakai.
Large-time asymptotics of the gyration radius for long-range statistical-mechanical models.
RIMS Kokyuroku Bessatsu B21 (2011): 53–62.

[6] R. van der Hofstad and A. Sakai.
Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: The higher-point functions.
Electron. J. Probab. 15 (2010): 801-894.

[7] L.-C. Chen and A. Sakai.
Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation.
Probab. Theory Relat. Fields 145 (2009): 435–458.

[8] A. Sakai.
Applications of the lace expansion to statistical-mechanical models.
A chapter in “Analysis and Stochastics of Growth Processes and Interface Models” (P. Möters et al. eds., Oxford University Press, 2008).

[9] M. Heydenreich, R. van der Hofstad and A. Sakai.
Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk.
J. Stat. Phys. 132 (2008): 1001–1049.

[10] L.-C. Chen and A. Sakai.
Critical behavior and the limit distribution for long-range oriented percolation. I.
Probab. Theory Relat. Fields 142 (2008): 151–188.

[11] M. Holmes and A. Sakai.
Senile reinforced random walks.
Stochastic Process. Appl. 117 (2007): 1519-1539.

[12] A. Sakai.
Lace expansion for the Ising model.
Commun. Math. Phys. 272 (2007): 283-344.

[13] 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.

[14] A. Sakai.
Mean-field behavior for the survival probability and the percolation point-to-surface connectivity.
J. Stat. Phys. 117 (2004): 111–130.

[15] R. van der Hofstad and A. Sakai.
Gaussian scaling for the critical spread-out contact process above the upper critical dimension.
Electron. J. Probab. 9 (2004): 710–769.

[16] M. Holmes, A.A. Járai, A. Sakai and G. Slade.
High-dimensional graphical networks of self-avoiding walks.
Canad. J. Math. 56 (2004): 77-114.

[17] A. Sakai.
Hyperscaling inequalities for the contact process and oriented percolation.
J. Stat. Phys. 106 (2002): 201–211.

[18] A. Sakai.
Mean-field critical behavior for the contact process.
J. Stat. Phys. 104 (2001): 111-143.

Keywords
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WebPage
http://www.math.sci.hokudai.ac.jp/~sakai/