Researcher Information



Challenge to the mystery of interacting many-body systems

Department of Mathematics, Mathematics


Rigorous analysis of phase transitions and critical behavior by using, e.g., the lace expansion; Evaluation of solutions to optimization problems by using the Ising model

FieldProbability, Statistical mechanics, Mathematical physics
KeywordPhase transitions, Critical phenomena, Optimization problems, Interacting many-body systems, The Ising model and other spin systems, Random walk and self-avoiding walk, Percolation, The lace expansion

Representative Achievements

Critical two-point function for long-range models with power-law couplings: The marginal case for d≧dc,
L.-C. Chen, A. Sakai,
Commun. Math. Phys., 372, 543–572 (2019).
Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees,
A. Sakai, G. Slade,
Electron. J. Probab., 24, no.65, 1–18 (2019).
Application of the Lace Expansion to the φ4 Model,
A. Sakai,
Commun. Math. Phys., 336, 619–648 (2015).
Mean-field Behavior for Long- and Finite Range Ising Model, Percolation and Self-avoiding Walk,
M. Heydenreich, R. van der Hofstad, A. Sakai,
J. Stat. Phys., 132, 1001–1049 (2008).
Lace Expansion for the Ising Model,
A. Sakai,
Commun. Math. Phys., 272, 283–344 (2007).
Affiliated academic societyThe Mathematical Society of Japan
ProjectCREST ``Steering Toward Spatio-Temporal Computing Architecture Driven by Learning/Math-Scientific Models"
JSPS KAKENHI ``Rigorous analysis for high-dimensional critical behavior and crossover phenomena in mathematical models"
Room addressScience Bldg #3 3-513

Department of Mathematics, Mathematics



What is the research theme that you are currently focusing on?

Mathematically rigorous analysis for phase transitions and critical behavior of many-body systems, such as sums of random variables and systems of interacting spin variables. The lace expansion is one of the few methods to deal with such problems in high dimensions. It is known that, if it is absolutely convergent, then the many-body effect is reduced and the critical behavior becomes simple. That is believed to occur only when the dimension d of the system is above the so-called upper-critical dimension dc, which is model-dependent. I have been trying to derive the lace expansion for other models (I became known for the lace expansion for the Ising model = a famous statistical mechanical model for magnets, and for the contact process = a stochastic process for the spread of a disease), to prove convergence of the lace expansion as soon as d > dc and to understand what happens when d =dc.

At the Seventh Wellington Workshop in Probability and Mathematical Statistics, December 2019
Please tell us your stories until you became a researcher.

It is essentially difficult to investigate phase transitions and critical behavior in a mathematically rigorous way, due to complicated interaction among infinitely many constituents. Relatively easy problems have already been solved, and the remaining ones are extremely challenging. Under this circumstance, and also because they are considered as problems in physics and chemistry, there has not been so much activity among Japanese mathematicians as compared to foreign researchers. This is one of the reasons why, after getting my Ph.D., I went abroad to work as a postdoctoral fellow at UBC (Canada), EURANDOM and TU/e (the Netherlands) and as a lecturer at the University of Bath in UK. During these 7 years or so, I have achieved a lot and gradually conquered anxiety about whether or not I can be a good researcher. I have also been able to establish my self-esteem, which was quite low when I started my academic career, and to eventually build another stronger personality. If I am asked to send a message to those who want to become academics, I would say: Do not worry too much, as anxiety can enhance your growth. There will always be good outcomes if you keep aiming high and continue working hard in the right direction!

Please tell us about yourself; things you are good at, your favorites, hobbies, and daily routines.

I believe I am good at doing researches (or bad at giving up so easily). Spending time on studying and doing researches is a major part of my daily routine. I should be thankful for this to be my occupation. Of course, it does not always proceed as it is initially planned. It is therefore necessary to be patient for a long period (the longest one was 7 and a half years for a single project). To maintain my mental capacity to carry out good researches, I also physically train every day.

With members of the L-Station