Researcher Database
| Akira Sakai |
| Faculty of Science Mathematics Mathematics |
| Professor |
Last Updated :2024/05/09
Researcher Profile and Settings
Affiliation
Job Title
Degree
URL
Research funding number
- 50506996
J-Global ID
Research Interests
- probability theory statistical mechanics mathematical physics phase transition critical behavior lace expansion Ising model φ^4 model contact process percolation lattice trees lattice animals self-avoiding walk random walk
Research Areas
- Natural sciences / Applied mathematics and statistics
- Natural sciences / Basic mathematics
- Natural sciences / Mathematical physics and basic theory
- Natural sciences / Mathematical analysis
Educational Organization
- Bachelor's degree program School of Science
- Master's degree program Graduate School of Science
- Doctoral (PhD) degree program Graduate School of Science
Academic & Professional Experience
- 2020/02 - Today Hokkaido University Faculty of Science Department of Mathematics Professor
- 2011/04 - 2020/01 Hokkaido University Faculty of Science Department of Mathematics Associate professor
- 2008/03 - 2011/03 Hokkaido University Creative Institution SOUSEI Tenure-track assistant professor
- 2006/04 - 2008/02 University of Bath Department of Mathematical Sciences Lecturer
- 2004/04 - 2006/03 Technische Universiteit Eindhoven Department of Mathematics and Computer Science Postdoctoral fellow
- 2003/01 - 2004/03 Eurandom Postdoctoral fellow
- 2001/01 - 2002/12 University of British Columbia Department of Mathematics Postdoctoral fellow
Education
- 1996/04 - 2000/12 Tokyo Institute of Technology
- 1994/04 - 1996/03 Tokyo Institute of Technology
- 1990/04 - 1994/03 Tokyo Institute of Technology
Association Memberships
Research Activities
Published Papers
- Work of Hugo Duminil-CopinAkira SakaiSugaku 76 (1) 48 - 60 2024/01 [Refereed][Invited]
- Bruno Hideki Fukushima-Kimura, Noe Kawamoto, Eitaro Noda, Akira SakaiJournal of Statistical Physics 190 (12) 2023/11/24 [Refereed]
- Noe Kawamoto, Akira SakaiCombinatorics, Probability and Computing 33 (2) 238 - 269 0963-5483 2023/11/20 [Refereed]
Abstract A spread-out lattice animal is a finite connected set of edges in $\{\{x,y\}\subset \mathbb{Z}^d\;:\;0\lt \|x-y\|\le L\}$. A lattice tree is a lattice animal with no loops. The best estimate on the critical point $p_{\textrm{c } }$ so far was achieved by Penrose (J. Stat. Phys. 77, 3–15, 1994) : $p_{\textrm{c } }=1/e+O(L^{-2d/7}\log L)$ for both models for all $d\ge 1$. In this paper, we show that $p_{\textrm{c } }=1/e+CL^{-d}+O(L^{-d-1})$ for all $d\gt 8$, where the model-dependent constant $C$ has the random-walk representation\begin{align*} C_{\textrm{LT } }=\sum _{n=2}^\infty \frac{n+1},{2e}U^{*n}(o),&& C_{\textrm{LA } }=C_{\textrm{LT } }-\frac 1{2e^2}\sum _{n=3}^\infty U^{*n}(o), \end{align*}where $U^{*n}$ is the $n$-fold convolution of the uniform distribution on the $d$-dimensional ball $\{x\in{\mathbb R}^d\;: \|x\|\le 1\}$. The proof is based on a novel use of the lace expansion for the 2-point function and detailed analysis of the 1-point function at a certain value of $p$ that is designed to make the analysis extremely simple. - Bruno Hideki Fukushima-Kimura, Satoshi Handa, Katsuhiro Kamakura, Yoshinori Kamijima, Kazushi Kawamura, Akira SakaiJournal of Statistical Physics 190 (4) 2023/03/22 [Refereed]
- Bruno Hideki Fukushima-Kimura, Yoshinori Kamijima, Kazushi Kawamura, Akira SakaiTransactions of the Institute of Systems, Control and Information Engineers 36 (1) 9 - 16 1342-5668 2023/01 [Refereed][Invited]
- Akira SakaiCommunications in Mathematical Physics 392 (3) 783 - 823 0010-3616 2022/06 [Refereed]
- Bruno Hideki Fukushima-Kimura, Yoshinori Kamijima, Kazushi Kawamura, Akira SakaiProceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications 2022 65 - 71 2188-4730 2022/03/31 [Refereed]
- Bruno Hideki Fukushima-Kimura, Akira Sakai, Hisayoshi Toyokawa, Yuki UedaPhysica A: Statistical Mechanics and its Applications 583 126208 - 126208 0378-4371 2021/12 [Refereed][Not invited]
- Kasho Yamamoto, Kazushi Kawamura, Kota Ando, Normann Mertig, Takashi Takemoto, Masanao Yamaoka, Hiroshi Teramoto, Akira Sakai, Shinya Takamaeda-Yamazaki, Masato MotomuraIEEE Journal of Solid-State Circuits 56 (1) 165 - 178 0018-9200 2021/01 [Refereed][Not invited]
- Satoshi Handa, Yoshinori Kamijima, Akira SakaiTaiwanese Journal of Mathematics 24 (3) 1027-5487 2020/06/01 [Refereed]
- Akira SakaiRIMS Kokyuroku Bessatsu B79 51 - 62 2020/04 [Refereed][Invited]
- Kasho Yamamoto, Kota Ando, Normann Mertig, Takashi Takemoto, Masanao Yamaoka, Hiroshi Teramoto, Akira Sakai, Shinya Takamaeda-Yamazaki, Masato Motomura2020 IEEE International Solid- State Circuits Conference(ISSCC) 138 - 140 2020 [Refereed][Not invited]
- Akira SakaiCommunications in Mathematical Physics 372 (2) 543 - 572 0010-3616 2019/12 [Refereed]
- Satoshi Handa, Markus Heydenreich, Akira SakaiSpringer Proceedings in Mathematics & Statistics 183 - 198 2194-1009 2019/10/18 [Refereed]
- Akira Sakai, Gordon SladeElectronic Journal of Probability 24 (none) 1083-6489 2019/01/01 [Refereed]
- Akira SakaiJOURNAL OF STATISTICAL PHYSICS 171 (3) 462 - 469 0022-4715 2018/05 [Refereed][Not invited]
Consider nearest-neighbor oriented percolation in space-time dimensions. Let be the critical exponents for the survival probability up to time t, the expected number of vertices at time t connected from the space-time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality , which holds for all and is a strict inequality above the upper-critical dimension 4, becomes an equality for , i.e., , provided existence of at least two among . The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin et al. [6]. - Toshihiro Arae, Shiori Isai, Akira Sakai, Katsuhiko Mineta, Masami Yokota Hirai, Yuya Suzuki, Shigehiko Kanaya, Junji Yamaguchi, Satoshi Naito, Yukako ChibaPLANT AND CELL PHYSIOLOGY 58 (6) 1090 - 1102 0032-0781 2017/06 [Refereed][Not invited]
Plants possess a cold acclimation system to acquire freezing tolerance through pre-exposure to non-freezing low temperatures. The transcriptional cascade of C-repeat-binding factors (CBFs)/dehydration response element-binding factors (DREBs) is considered a major transcriptional regulatory pathway during cold acclimation. However, little is known regarding the functional significance of mRNA stability regulation in the response of gene expression to cold stress. The actual level of individual mRNAs is determined by a balance between mRNA synthesis and degradation. Therefore, it is important to assess the regulatory steps to increase our understanding of gene regulation. Here, we analyzed temporal changes in mRNA amounts and half-lives in response to cold stress in Arabidopsis cell cultures based on genome-wide analysis. In this mRNA decay array method, mRNA half-life measurements and microarray analyses were combined. In addition, temporal changes in the integrated value of transcription rates were estimated from the above two parameters using a mathematical approach. Our results showed that several cold-responsive genes, including Cold-regulated 15a, were relatively destabilized, whereas the mRNA amounts were increased during cold treatment by accelerating the transcription rate to overcome the destabilization. Considering the kinetics of mRNA synthesis and degradation, this apparently contradictory result supports that mRNA destabilization is advantageous for the swift increase in CBF-responsive genes in response to cold stress. - Yuki Chino, Akira SakaiJOURNAL OF STATISTICAL PHYSICS 163 (4) 754 - 764 0022-4715 2016/05 [Refereed][Not invited]
Following similar analysis to that in Lacoin (Probab Theory Relat Fields 159: 777-808, 2014), we can show that the quenched critical point for self-avoiding walk on random conductors on Z(d) is almost surely a constant, which does not depend on the location of the reference point. We provide upper and lower bounds which are valid for all d >= 1. - Akira SakaiCOMMUNICATIONS IN MATHEMATICAL PHYSICS 336 (2) 619 - 648 0010-3616 2015/06 [Refereed][Not invited]
Using the Griffiths-Simon construction of the model and the lace expansion for the Ising model, we prove that, if the strength of nonlinearity is sufficiently small for a large class of short-range models in dimensions d > 4, then the critical two-point function is asymptotically times a model-dependent constant, and the critical point is estimated as , where is the massless point for the Gaussian model. - Lung-Chi Chen, Akira SakaiANNALS OF PROBABILITY 43 (2) 639 - 681 0091-1798 2015/03 [Refereed][Not invited]
We consider long-range self-avoiding walk, percolation and the Ising model on Z(d) that are defined by power-law decaying pair potentials of the form D(x) asymptotic to vertical bar x vertical bar(-d-alpha) with alpha > 0. The upper-critical dimension d(c) is 2(alpha boolean AND 2) for self-avoiding walk and the Ising model, and 3(alpha boolean AND 2) for percolation. Let alpha not equal 2 and assume certain heat-kernel bounds on the n-step distribution of the underlying random walk. We prove that, for d > d(c) (and the spread-out parameter sufficiently large), the critical two-point function G p(c) (X) for each model is asymptotically C vertical bar x vertical bar(alpha boolean AND 2-d), where the constant C is an element of (0, infinity) is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between alpha < 2 and alpha > 2. We also provide a class of random walks that satisfy those heat-kernel bounds. - Lung-Chi Chen, Akira SakaiANNALS OF PROBABILITY 39 (2) 507 - 548 0091-1798 2011/03 [Refereed][Not invited]
We consider random walk and self-avoiding walk whose 1-step distribution is given by D, and oriented percolation whose bond-occupation probability is proportional to D. Suppose that D(x) decays as vertical bar x vertical bar(-d-alpha) with alpha > 0. For random walk in any dimension d and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension d(c) equivalent to 2(alpha boolean AND 2), we prove large-t asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length t or the average spatial size of an oriented percolation cluster at time t. This proves the conjecture for long-range self-avoiding walk in [Ann. Inst. H. Poincare Probab. Statist. (2010), to appear] and for long-range oriented percolation in [Probab. Theory Related Fields 142 (2008) 151-188] and [Probab. Theory Related Fields 145 (2009) 435-458]. - Akira SakaiRIMS Kokyuroku Bessatsu 京都大学 B21 53 - 61 1881-6193 2010/12 [Not refereed][Not invited]
- Remco van der Hofstad, Akira SakaiELECTRONIC JOURNAL OF PROBABILITY 15 801 - 894 1083-6489 2010/06 [Refereed][Not invited]
We consider the critical spread-out contact process in Z(d) with d >= 1, whose infection range is denoted by L >= 1. In this paper, we investigate the higher-point functions tau((r))((t) over right arrow)((x) over right arrow) for r >= 3, where tau((r))((t) over right arrow)((x) over right arrow) is the probability that, for all i = 1,...,r-1, the individual located at x(i) is an element of Z(d) is infected at time t(i) by the individual at the origin o is an element of Z(d) at time 0. Together with the results of the 2-point function in [16], on which our proofs crucially rely, we prove that the r-point functions converge to the moment measures of the canonical measure of super-Brownian motion above the upper critical dimension 4. We also prove partial results for d <= 4 in a local mean-field setting. The proof is based on the lace expansion for the time-discretized contact process, which is a version of oriented percolation in Z(d) x epsilon Z(+), where epsilon is an element of (0,1] is the time unit. For ordinary oriented percolation (i.e., epsilon = 1), we thus reprove the results of [20]. The lace expansion coefficients are shown to obey bounds uniformly in epsilon is an element of (0,1], which allows us to establish the scaling results also for the contact process (i.e., epsilon down arrow 0). We also show that the main term of the vertex factor V, which is one of the non-universal constants in the scaling limit, is 2 - epsilon (= 1 for oriented percolation, = 2 for the contact process), while the main terms of the other non-universal constants are independent of epsilon. The lace expansion we develop in this paper is adapted to both the r-point function and the survival probability. This unified approach makes it easier to relate the expansion coefficients derived in this paper and the expansion coefficients for the survival probability, which will be investigated in [18]. - Lung-Chi Chen, Akira SakaiPROBABILITY THEORY AND RELATED FIELDS 145 (3-4) 435 - 458 0178-8051 2009/11 [Refereed][Not invited]
We prove that the Fourier transform of the properly scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index alpha > 0 converges to e(-C)vertical bar k vertical bar(alpha boolean AND 2) for some C is an element of (0, infinity) above the upper- critical dimension d(c) equivalent to 2(alpha boolean AND 2). This answers the open question remained in the previous paper (Chen and Sakai in Probab Theory Relat Fields 142:151-188, 2008). Moreover, we show that the constant C exhibits crossover at alpha = 2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients. - Markus Heydenreich, Remco van der Hofstad, Akira SakaiJOURNAL OF STATISTICAL PHYSICS 132 (6) 1001 - 1049 0022-4715 2008/09 [Refereed][Not invited]
We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if d > 2(alpha boolean AND 2) for self-avoiding walk and the Ising model, and d > 3(alpha boolean AND 2) for percolation, where d denotes the dimension and alpha the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (Ann. Probab. 33(5):1886-1944, 2005). - Lung-Chi Chen, Akira SakaiPROBABILITY THEORY AND RELATED FIELDS 142 (1-2) 151 - 188 0178-8051 2008/09 [Refereed][Not invited]
We consider oriented percolation on Z(d) x Z(+) whose bond-occupation probability is pD(center dot), where p is the percolation parameter and D is a probability distribution on Z(d). Suppose that D(x) decays as vertical bar x vertical bar(-d-alpha) for some alpha > 0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension d(c) = 2(alpha boolean AND 2). We also show that, for every k, the Fourier transform of the normalized two-point function at time n, with a proper spatial scaling, has a convergent subsequence to e-c vertical bar k vertical bar(alpha boolean AND 2) for some c > 0. - Akira SakaiAnalysis and Stochastics of Growth Processes and Interface Models Oxford University Press 3 123 - 147 2008/07/24 [Refereed][Invited]
Synergetics is a common feature in interesting statistical-mechanical problems. One of the most important examples of synergetics is the emergence of a second-order phase transition and critical behaviour. It is rich and still far from fully understood. The reason why it is so difficult is due to the increase to infinity of the number of strongly correlated variables in the vicinity of the critical point. For example, the Ising model, which is a model for magnets, exhibits critical behaviour as the temperature comes closer to its critical value; the closer the temperature is to criticality, the more spin variables cooperate with each other to attain the global magnetization. In this regime, neither standard probability theory for independent random variables nor naive perturbation techniques work. The lace expansion, which is the topic of this article, is currently one of the few approaches to rigorous investigation of critical behaviour for various statistical-mechanical models. The chapter summarizes some of the most intriguing lace-expansion results for self-avoiding walk (SAW), percolation, and the Ising model.Analysis and stochastics of growth processes and interface models - M. Holmes, A. SakaiSTOCHASTIC PROCESSES AND THEIR APPLICATIONS 117 (10) 1519 - 1539 0304-4149 2007/10 [Refereed][Not invited]
We consider random walks with transition probabilities depending on the number of consecutive traversals n of the edge most recently traversed. Such walks may get stuck on a single edge, or have every vertex recurrent or every vertex transient, depending on the reinforcement function f (n) that characterizes the model. We prove recurrence/transience results when the walk does not get stuck on a single edge. We also show that the diffusion constant need not be monotone in the reinforcement. (C) 2007 Elsevier B.V. All rights reserved. - Akira Sakai2007/08/21 [Not refereed][Not invited]
We provide a complete proof of the diagrammatic bounds on the lace-expansion coefficients for oriented percolation, which are used in [arXiv:math/0703455 ] to investigate critical behavior for long-range oriented percolation above 2\min{\alpha,2} spatial dimensions. - Akira SakaiCOMMUNICATIONS IN MATHEMATICAL PHYSICS 272 (2) 283 - 344 0010-3616 2007/06 [Refereed][Not invited]
The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk. As a result, we obtain Gaussian asymptotics of the critical two-point function for the nearest-neighbor model with d >> 4 and for the spread-out model with d > 4 and L >> 1, without assuming reflection positivity. - R van der Hofstad, A SakaiPROBABILITY THEORY AND RELATED FIELDS 132 (3) 438 - 470 0178-8051 2005/07 [Refereed][Not invited]
We consider self-avoiding walk and percolation in Z(d), oriented percolation in X-d x Z(+), and the contact process in Z(d), with pD(center dot) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently occupied with probability p D(y-x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical point p(c). We investigate the value of p(c) when d > 6 for percolation and d > 4 for the other models, and L >> 1. We prove in a unified way that p(c)=1+C(D)+O(L-2), where the universal term 1 is the mean-field critical value, and the model-dependent term C(D)=O(L-d) is written explicitly in terms of the random walk transition probability D. We also use this result to prove that p(c)=1+cL(-d) +O(L-d-1), where c is a model-dependent constant. Our proof is based on the lace expansion for each of these models. - Akira SakaiJOURNAL OF STATISTICAL PHYSICS 117 (1-2) 111 - 130 0022-4715 2004/10 [Refereed][Not invited]
We consider the critical survival probability (up to time t) for oriented percolation and the contact process, and the point-to-surface (of the ball of radius t) connectivity for critical percolation. Let theta(t) denote both quantities. We prove in a unified fashion that, if theta(t) exhibits a power law and both the two-point function and its certain restricted version exhibit the same mean-field behavior, then theta(t) asymptotic to t(-1) for the time-oriented models with d > 4 and theta(t) asymptotic to t(-2) for percolation with d > 7. - R van der Hofstad, A SakaiELECTRONIC JOURNAL OF PROBABILITY 9 710 - 769 1083-6489 2004/10 [Refereed][Not invited]
We consider the critical spread-out contact process in Z(d) with d greater than or equal to 1, whose infection range is denoted by L greater than or equal to 1. The two-point function tau(t)(x) is the probability that x is an element of Z(d) is infected at time t by the infected individual located at the origin o is an element of Z(d) at time 0. We prove Gaussian behaviour for the two-point function with L greater than or equal to L(o) for some finite L(o) = L(o)(d) for d > 4. When d less than or equal to 4, we also perform a local mean-field limit to obtain Gaussian behaviour for tau(tT) (x) with t > 0 fixed and T --> infinity when the infection range depends on T in such a way that L(T) = LT(b) for any b > (4 - d)/2d. The proof is based on the lace expansion and an adaptation of the inductive approach applied to the discretized contact process. We prove the existence of several critical exponents and show that they take on their respective mean-field values. The results in this paper provide crucial ingredients to prove convergence of the finite-dimensional distributions for the contact process towards those for the canonical measure of super-Brownian motion, which we defer to a sequel of this paper. The results in this paper also apply to oriented percolation, for which we reprove some of the results in [20] and extend the results to the local mean-field setting described above when d less than or equal to 4. - M Holmes, AA Jarai, A Sakai, G SladeCANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES 56 (1) 77 - 114 0008-414X 2004/02 [Refereed][Not invited]
We use the lace expansion to analyse networks of mutually-avoiding self-avoiding walks, having the topology of a graph. The networks are defined in terms of spread-out self-avoiding walks that are permitted to take large steps. We study the asymptotic behaviour of networks in the limit of widely separated network branch points, and prove Gaussian behaviour for sufficiently spread-out networks on Z(d) in dimensions d > 4. - Akira SakaiJOURNAL OF STATISTICAL PHYSICS 106 (1-2) 201 - 211 0022-4715 2002/01 [Refereed][Not invited]
The contact process and oriented percolation are expected to exhibit the same critical behavior in any dimension. Above their upper critical dimension d(c) they exhibit the same critical behavior as the branching process. Assuming existence of the critical exponents, we prove a pair of hyperscaling inequalities which, together with the results of refs. 16 and 18, implies d(c) = 4. - Akira SakaiJOURNAL OF STATISTICAL PHYSICS 104 (1-2) 111 - 143 0022-4715 2001/07 [Refereed][Not invited]
The contact process is a model of spread of an infectious disease. Combining with the result of ref. 1, we prove that the critical exponents take on the mean-field values for sufficiently high dimensional nearest-neighbor models and for sufficiently spread-out models with d > 4: theta(lambda) approximate to lambda - lambda (c) as lambda down arrow lambda (c) and chi(lambda) approximate to (lambda (c) - lambda)(-1) as lambda up arrow lambda where theta(lambda) and chi(lambda) are the spread probability and the susceptibility of the infection respectively, and lambda (c) is the critical infection rate. Our results imply that the upper critical dimension for the contact process is at most 4.
Books etc
- Satoshi Handa, Markus Heydenreich, Akira Sakai (ContributorMean-field bound on the 1-arm exponent for Ising ferromagnets in high dimensions)
Springer Singapore 2019 (ISBN: 9789811502941) 183-198 - Analysis and Stochastics of Growth Processes and Interface ModelsAkira Sakai (ContributorApplications of the lace expansion to statistical-mechanical models)
Oxford University Press 2008/07 (ISBN: 9780199239252) 352 123-147
MISC
- 上島 芳倫, 坂井 哲 数理解析研究所講究録 (2246) 5 -15 2023/04
- 上島 芳倫, 坂井 哲, 半田 悟 数理解析研究所講究録 (2030) 135 -142 2017/05
- Sakai Akira RIMS Kokyuroku 1855- 80 -81 2013/10
Awards & Honors
- 2013/03 北海道大学 The Hokkaido University President’s Award for Teaching Excellence in 2013
受賞者: 坂井哲 - 2012/03 北海道大学 The Hokkaido University President’s Award for Teaching Excellence in 2012
受賞者: 坂井哲
Research Grants & Projects
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2023/04 -2027/03Author : 宮尾 忠宏, 松井 卓, 坂井 哲, 桂 法称, 松澤 泰道
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2023/04 -2026/03Author : 坂井 哲
- Japan Science and Technology Agency：Strategic Basic Research Programs CRESTDate (from‐to) : 2018/10 -2024/03Author : Akira Sakai
- Ministry of Education, Culture, Sports, Science and Technology - Japan：Grant-in-Aid for Scientific Research(基盤研究(C))Date (from‐to) : 2018/04 -2022/03Author : Akira Sakai本研究では，強磁性イジング模型や自己回避歩行，パーコレーションなどの数理モデルの 相転移・臨界現象に関する次の3つの未解決問題を研究する．（1）d次元体心立方格子上のパーコレーションがd≧7で平均場臨界現象を示すことを証明し，最近接パーコレーションの上部臨界次元が6であるという予想を解決すること．（2）2体相互作用係数が2体間距離の冪で減衰する長距離モデルに対し，特にその分散が対数発散する境界冪の場合，上部臨界次元以上（等号込）で臨界2点関数の漸近挙動を解明すること．（3）並進対称性の成り立たない「ランダムな媒質中の自己回避歩行」を解析する手法を確立し，十分高次元では均質な媒質中の自己回避歩行と同じ臨界現象を示すという予想を解決すること． （1）d≧9で平均場臨界現象への退化を証明し，論文はTaiwanese Journal of Mathematics（2020年）に掲載された．これは，標準的なd次元正方格子上における上部臨界次元の評価d≧11（Fitznerとvan der Hofstadによる2017年の結果）を凌駕している． （2）対数発散のないα≠2の結果はAnnals of Probability（2015年）に掲載されていたが，この度，最も難しいα＝2の場合も解決し，Communications in Mathematical Physics（2019年）と 数理解析研究所講究録別冊（2020年）に論文が掲載された． （3）コロナ下でお互い密に情報交換できる場が設けられず，思うような進展は見られていない．
- Ministry of Education, Culture, Sports, Science and Technology - Japan：Grant-in-Aid for Challenging Exploratory ResearchDate (from‐to) : 2015/04 -2018/03Author : Akira SakaiConsider the 1-spin expectation at the center of the d-dimensional ball of radius r under the plus-boundary condition. It has been known that it exhibits a phase transition and decays to zero as r diverges at the critical temperature. In particular, it is known to decay slower than r to the power 1-d/2 above the upper-critical dimension 4, due to a hyperscaling inequality. By using the random-current representation, a stochastic-geometric representation for the Ising model, Handa, Heydenreich and I have proven a sharper second-moment estimate and concluded that the critical 1-spin expectation decays no faster than 1/r above 4 dimensions, meaning the 1-arm exponent is not bigger than the long-expected mean-field value 1 for d>4. The results are summarized in a paper, which was accepted for publication in a festschrift for Charles Newman's 70th birthday.
- Ministry of Education, Culture, Sports, Science and Technology - Japan：Grants-in-Aid for Scientific Research(基盤研究(C))Date (from‐to) : 2012/04 -2015/03Author : Akira SakaiThe (ferromagnatic) Ising model and the φ4 model are known to exhibit phase transition and critical behavior. In 2007, Sakai used a stochastic-geometrical representation, known as the random-current representation, to develop the lace expansion for the Ising model. Extending the use of this stochastic-geometrical representation, we applied the lace expansion to the φ4 model and obtained an asymptotic expression of the critical two-point function in high dimensions. We also established the method of analyzing critical behavior for the models defined by power-law decaying pair potentials, and proved that the critical two-point function in high dimensions is asymptotically Newtonian or Riesz, depending on the value of the power exponent of the pair potentials.
- Ministry of Education, Culture, Sports, Science and Technology：Grants-in-Aid for Scientific Research(若手研究(B))Date (from‐to) : 2009/04 -2012/03Author : Akira SAKAIThe lace expansion has been one of the few mathematically rigorous approaches to investigate critical behavior in high dimensions. We have extended this methodology to obtain a universal sharp asymptotic expression of the 2-point functions for long-range self-avoiding walk and long-range oriented percolation which are defined by power-law decaying pair potentials. We have also investigated the finite-range(but sufficiently spread-out) critical contact process and proved that the n-point function under the Brownian scaling converges to the(n-1)-point function for the canonical measure of super-Brownian motion.
Educational Activities
Teaching Experience
- The World of Science and Technology開講年度 : 2021課程区分 : 学士課程開講学部 : 全学教育キーワード : 1. 最小二乗法, 正規分布，統計的推測（小林） 2. マルコフ過程，定常分布，固有値，固有ベクトル (坂井) 3. 最適化，ラグランジュの未定乗数法，フレシュ微分，形状最適化 (正宗) 4. 素因数分解, 群論, 暗号理論 (朝倉)
- Calculus II開講年度 : 2021課程区分 : 学士課程開講学部 : 全学教育キーワード : 原始関数, 積分, 重積分, リ－マン和, 変数変換
- Introduction to statistics開講年度 : 2021課程区分 : 学士課程開講学部 : 理学部キーワード : （条件つき）確率，独立性，確率変数，平均，（共）分散，大数の法則，中心極限定理，推定，検定
- Basic Mathematical Science開講年度 : 2021課程区分 : 学士課程開講学部 : 理学部キーワード : 坂井： 電気回路，調和関数，ランダムウォーク，マルチンゲール 行木： 離散力学系、軌道の安定性、分岐現象、非線形時系列解析