SAKAI, Akira
 Research Field
 Applied Mathematics
 Position
 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 statisticalmechanical 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 φ^{4} model (in lattice scalarfield theory),
* selfavoiding 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 selfavoiding walk on random conductors.
J. Stat. Phys. 163 (2016): 754764.[2] A. Sakai.
Application of the lace expansion to the φ^{4} model.
Commun. Math. Phys. 336 (2015): 619648.[3] L.C. Chen and A. Sakai.
Critical twopoint functions for longrange statisticalmechanical models in high dimensions.
Ann. Probab. 43 (2015): 639681.[4] L.C. Chen and A. Sakai.
Asymptotic behavior of the gyration radius for longrange selfavoiding walk and longrange oriented percolation.
Ann. Probab. 39 (2011) 507548[5] A. Sakai.
Largetime asymptotics of the gyration radius for longrange statisticalmechanical models.
RIMS Kokyuroku Bessatsu B21 (2011): 53–62.[6] R. van der Hofstad and A. Sakai.
Convergence of the critical finiterange contact process to superBrownian motion above the upper critical dimension: The higherpoint functions.
Electron. J. Probab. 15 (2010): 801894.[7] L.C. Chen and A. Sakai.
Critical behavior and the limit distribution for longrange oriented percolation. II: Spatial correlation.
Probab. Theory Relat. Fields 145 (2009): 435–458.[8] A. Sakai.
Applications of the lace expansion to statisticalmechanical 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.
Meanfield behavior for long and finite range Ising model, percolation and selfavoiding walk.
J. Stat. Phys. 132 (2008): 1001–1049.[10] L.C. Chen and A. Sakai.
Critical behavior and the limit distribution for longrange 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): 15191539.[12] A. Sakai.
Lace expansion for the Ising model.
Commun. Math. Phys. 272 (2007): 283344.[13] R. van der Hofstad and A. Sakai.
Critical points for spreadout selfavoiding walk, percolation and the contact process above the upper critical dimensions.
Probab. Theory Relat. Fields 132 (2005): 438470.[14] A. Sakai.
Meanfield behavior for the survival probability and the percolation pointtosurface connectivity.
J. Stat. Phys. 117 (2004): 111–130.[15] R. van der Hofstad and A. Sakai.
Gaussian scaling for the critical spreadout 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.
Highdimensional graphical networks of selfavoiding walks.
Canad. J. Math. 56 (2004): 77114.[17] A. Sakai.
Hyperscaling inequalities for the contact process and oriented percolation.
J. Stat. Phys. 106 (2002): 201–211.[18] A. Sakai.
Meanfield critical behavior for the contact process.
J. Stat. Phys. 104 (2001): 111143.  Keywords
 Critical phenomena, Interacting particle systems, IsingPotts model, Lace expansion, Percolation, Phase transitions
 WebPage
 http://www.math.sci.hokudai.ac.jp/~sakai/