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研究者情報

マスター

アカウント（マスター）

  • 氏名

    坂井　哲(サカイ　アキラ), サカイ　アキラ

所属（マスター）

  • 理学研究院　数学部門　数学分野

所属（マスター）

  • 理学研究院　数学部門　数学分野

独自項目

syllabus

  • 2021, 科学・技術の世界, The World of Science and Technology, 学士課程, 全学教育, 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)
  • 2021, 微分積分学Ⅱ, Calculus II, 学士課程, 全学教育, 原始関数, 積分, 重積分, リ－マン和, 変数変換
  • 2021, 確率・統計入門, Introduction to statistics, 学士課程, 理学部, （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
  • 2021, 数理科学基礎, Basic Mathematical Science, 学士課程, 理学部, 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

researchmap

プロフィール情報

学位

  • 博士（理学）（東京工業大学）

プロフィール情報

  • 姓

    坂井, サカイ

  • 名

    哲, アキラ

  • ID各種

    201301005315886905

対象リソース

業績リスト

研究キーワード

  • 確率論   統計力学   数理物理   相転移   臨界現象   レース展開   イジング模型   φ^4模型   コンタクトプロセス   パーコレーション   格子樹   格子動物   自己回避歩行   ランダムウォーク   

研究分野

  • 自然科学一般 / 応用数学、統計数学
  • 自然科学一般 / 数学基礎
  • 自然科学一般 / 数理物理、物性基礎
  • 自然科学一般 / 数理解析学

経歴

  • 2020年02月 - 現在 北海道大学 大学院理学研究院 数学部門 教授
  • 2011年04月 - 2020年01月 北海道大学 大学院理学研究院 数学部門 准教授
  • 2008年03月 - 2011年03月 北海道大学 創成研究機構 特任助教
  • 2006年04月 - 2008年02月 バース大学 数理科学科 講師
  • 2004年04月 - 2006年03月 アイントホーフェン工科大学 数学・コンピュータ科学科 ポスドク
  • 2003年01月 - 2004年03月 ユーランダム ポスドク
  • 2001年01月 - 2002年12月 ブリティッシュコロンビア大学 数学科 ポスドク

学歴

  • 1996年04月 - 2000年12月   東京工業大学大学院
  • 1994年04月 - 1996年03月   東京工業大学大学院
  • 1990年04月 - 1994年03月   東京工業大学

委員歴

  • 2020年03月 - 現在   Mathematical Physics, Analysis and Geometry   准編集者
  • 2020年08月 - 2023年10月   Taiwanese Journal of Mathematics   准編集者
  • 2020年04月 - 2022年03月   日本数学会   統計数学分科会運営委員
  • 2015年09月 - 2019年08月   Bernoulli Society   評議員

受賞

  • 2013年03月 北海道大学 平成25年度 北海道大学教育総長賞
     
    受賞者: 坂井哲
  • 2012年03月 北海道大学 平成24年度 北海道大学教育総長賞
     
    受賞者: 坂井哲

論文

  • Hugo Duminil-Copin氏の業績
    坂井 哲
    数学 76 1 48 - 60 2024年01月 [査読有り][招待有り]
  • Bruno Hideki Fukushima-Kimura, Noe Kawamoto, Eitaro Noda, Akira Sakai
    Journal of Statistical Physics 190 12 2023年11月24日 [査読有り]
  • Noe Kawamoto, Akira Sakai
    Combinatorics, Probability and Computing 33 2 238 - 269 2023年11月20日 [査読有り]
     
    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 Sakai
    Journal of Statistical Physics 190 4 2023年03月22日 [査読有り]
  • Bruno Hideki Fukushima-Kimura, Yoshinori Kamijima, Kazushi Kawamura, Akira Sakai
    Transactions of the Institute of Systems, Control and Information Engineers 36 1 9 - 16 2023年01月 [査読有り][招待有り]
  • 坂井 哲
    数学 74 3 253 - 279 2022年07月 [査読有り][招待有り]
  • Akira Sakai
    Communications in Mathematical Physics 392 3 783 - 823 2022年06月 [査読有り]
  • Bruno Hideki Fukushima-Kimura, Yoshinori Kamijima, Kazushi Kawamura, Akira Sakai
    Proceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications 2022 65 - 71 2022年03月31日 [査読有り]
  • Bruno Hideki Fukushima-Kimura, Akira Sakai, Hisayoshi Toyokawa, Yuki Ueda
    Physica A: Statistical Mechanics and its Applications 583 126208 - 126208 2021年12月 [査読有り][通常論文]
  • Kasho Yamamoto, Kazushi Kawamura, Kota Ando, Normann Mertig, Takashi Takemoto, Masanao Yamaoka, Hiroshi Teramoto, Akira Sakai, Shinya Takamaeda-Yamazaki, Masato Motomura
    IEEE Journal of Solid-State Circuits 56 1 165 - 178 2021年01月 [査読有り][通常論文]
  • Satoshi Handa, Yoshinori Kamijima, Akira Sakai
    Taiwanese Journal of Mathematics 24 3 2020年06月01日 [査読有り]
  • Akira Sakai
    RIMS Kokyuroku Bessatsu B79 51 - 62 2020年04月 [査読有り][招待有り]
  • Kasho Yamamoto, Kota Ando, Normann Mertig, Takashi Takemoto, Masanao Yamaoka, Hiroshi Teramoto, Akira Sakai, Shinya Takamaeda-Yamazaki, Masato Motomura
    2020 IEEE International Solid- State Circuits Conference(ISSCC) 138 - 140 2020年 [査読有り][通常論文]
  • Lung-Chi Chen, Akira Sakai
    Communications in Mathematical Physics 372 2 543 - 572 2019年12月 [査読有り]
  • Satoshi Handa, Markus Heydenreich, Akira Sakai
    Springer Proceedings in Mathematics & Statistics 183 - 198 2019年10月18日 [査読有り]
  • Akira Sakai, Gordon Slade
    Electronic Journal of Probability 24 none 2019年01月01日 [査読有り]
  • Akira Sakai
    JOURNAL OF STATISTICAL PHYSICS 171 3 462 - 469 2018年05月 [査読有り][通常論文]
     
    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 Chiba
    PLANT AND CELL PHYSIOLOGY 58 6 1090 - 1102 2017年06月 [査読有り][通常論文]
     
    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 Sakai
    JOURNAL OF STATISTICAL PHYSICS 163 4 754 - 764 2016年05月 [査読有り][通常論文]
     
    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 Sakai
    Communications in Mathematical Physics 336 2 619 - 648 2015年06月 [査読有り][通常論文]
     
    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 Sakai
    ANNALS OF PROBABILITY 43 2 639 - 681 2015年03月 [査読有り][通常論文]
     
    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 Sakai
    ANNALS OF PROBABILITY 39 2 507 - 548 2011年03月 [査読有り][通常論文]
     
    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 Sakai
    RIMS Kokyuroku Bessatsu B21 53 - 61 京都大学 2010年12月 [査読無し][通常論文]
  • Remco van der Hofstad, Akira Sakai
    ELECTRONIC JOURNAL OF PROBABILITY 15 801 - 894 2010年06月 [査読有り][通常論文]
     
    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 Sakai
    PROBABILITY THEORY AND RELATED FIELDS 145 3-4 435 - 458 2009年11月 [査読有り][通常論文]
     
    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 Sakai
    JOURNAL OF STATISTICAL PHYSICS 132 6 1001 - 1049 2008年09月 [査読有り][通常論文]
     
    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 Sakai
    PROBABILITY THEORY AND RELATED FIELDS 142 1-2 151 - 188 2008年09月 [査読有り][通常論文]
     
    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.
  • Sakai Akira
    Analysis and Stochastics of Growth Processes and Interface Models 3 123 - 147 Oxford University Press 2008年07月24日 [査読有り][招待有り]
     
    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. Sakai
    STOCHASTIC PROCESSES AND THEIR APPLICATIONS 117 10 1519 - 1539 2007年10月 [査読有り][通常論文]
     
    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 Sakai
    2007年08月21日 [査読無し][通常論文]
     
    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 Sakai
    COMMUNICATIONS IN MATHEMATICAL PHYSICS 272 2 283 - 344 2007年06月 [査読有り][通常論文]
     
    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 Sakai
    PROBABILITY THEORY AND RELATED FIELDS 132 3 438 - 470 2005年07月 [査読有り][通常論文]
     
    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 Sakai
    JOURNAL OF STATISTICAL PHYSICS 117 1-2 111 - 130 2004年10月 [査読有り][通常論文]
     
    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 Sakai
    ELECTRONIC JOURNAL OF PROBABILITY 9 710 - 769 2004年10月 [査読有り][通常論文]
     
    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 Slade
    CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES 56 1 77 - 114 2004年02月 [査読有り][通常論文]
     
    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 Sakai
    JOURNAL OF STATISTICAL PHYSICS 106 1-2 201 - 211 2002年01月 [査読有り][通常論文]
     
    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 Sakai
    JOURNAL OF STATISTICAL PHYSICS 104 1-2 111 - 143 2001年07月 [査読有り][通常論文]
     
    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.

MISC

書籍等出版物

  • Satoshi Handa, Markus Heydenreich, Akira Sakai (担当:分担執筆範囲:Mean-field bound on the 1-arm exponent for Ising ferromagnets in high dimensions)
    Springer Singapore 2019年 (ISBN: 9789811502941) 183-198
  • Analysis and Stochastics of Growth Processes and Interface Models
    Akira Sakai (担当:分担執筆範囲:Applications of the lace expansion to statistical-mechanical models)
    Oxford University Press 2008年07月 (ISBN: 9780199239252) 352 123-147

所属学協会

  • 日本数学会   システム制御情報学会   

共同研究・競争的資金等の研究課題

  • 日本学術振興会：科学研究費助成事業
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  • 日本学術振興会：科学研究費助成事業
    研究期間 : 2023年04月 -2026年03月 
    代表者 : 坂井 哲
  • 独立行政法人科学技術振興機構：戦略的創造研究推進事業CREST
    研究期間 : 2018年10月 -2024年03月 
    代表者 : 坂井 哲
  • 文部科学省：科学研究費補助金(基盤研究(C))
    研究期間 : 2018年04月 -2022年03月 
    代表者 : 坂井哲
     
    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．
  • 文部科学省：科学研究費補助金(挑戦的萌芽研究)
    研究期間 : 2015年04月 -2018年03月 
    代表者 : 坂井哲
     
    半径ｒのｄ次元球をプラス境界条件下におき，その中心のスピンの期待値を考える．この１スピン期待値は相転移を示し，臨界温度直上ではｒ→∞でゼロに収束することが知られていた．とくに上部臨界次元である４次元より上では，ｒの（ｄ－２）／２乗の逆数より遅く減衰することが，ハイパースケーリング不等式から厳密に知られていた． 半田氏とHeydenreich氏との共同研究において，確率幾何的表現である「ランダムカレント表示」を用いて２次モーメント法を精密に評価し，臨界１スピン期待値が４次元より上では１／ｒよりもゆっくり減衰することを厳密に証明した．Newman氏のfestschriftに掲載されることが決定．
  • 文部科学省：科学研究費補助金(基盤研究(C))
    研究期間 : 2012年04月 -2015年03月 
    代表者 : 坂井 哲
     
    スピン系数理モデルの臨界現象を「確率幾何学的な解析」によって厳密に理解するべく，以下の3つの課題に取り組んでいる．（１）冪的に減衰する長距離２体相互作用で定義されたイジング模型を考え，その冪指数（＝ｄ＋α）に依存して，高次元における臨界２点関数が漸近的にリース核になったり（α＜２），ニュートン核になったり（α＞２），それに対数補正がつく（α＝２）という予想を解決．（２）φ４乗モデルに対するレース展開の手法を確立し，「繰り込み群」による従来の結果を凌駕する精密な結果を導出．（３）「＋境界条件」が課されたｄ次元球の中心に位置するイジングスピンの期待値が，球の半径の或る冪（この冪指数を１-arm指数と呼び，ｄ＞４の最近接格子モデルでは１）で減衰するという予想の解決．平成24年度のそれぞれの課題に対する実績は次の通り．（１）8月に共同研究者のChen教授が訪日，11月と3月に坂井が訪台し，α≠２の結果をまとめて論文を執筆，確率論トップジャーナル“Annals of Probability”に投稿，受理された．α＝２については現在進行中．（２）現在までに得られている解析結果を基に，Universita di Modena e Reggio Emilia（イタリア，6月），Mathematisches Forschungsinstitut Oberwolfach（ドイツ，9月），National Center for Theoretical Sciences（台湾，3月）で開催された国際研究集会で招待講演を行ない，確率論シンポジウム（京都，12月）で発表．論文は現在執筆中．（３）共同研究者のHeydenreich教授とMathematisches Forschungsinstitut Oberwolfach（ドイツ，9月）で研究打ち合わせを行ない，課題が山積していることを再確認した．
  • 文部科学省：科学研究費補助金(若手研究(B))
    研究期間 : 2009年04月 -2012年03月 
    代表者 : 坂井 哲
     
    高次元臨界現象を数学的に厳密に解析できる数少ない(モデルによっては殆ど唯一の)手段に「レース展開」がある.この強力な解析手法を発展させ, 2体相互作用係数が距離の冪で減衰する自己回避歩行や有向パーコレーションの2点関数の漸近挙動を精密に求めることに成功した.また, 2体相互作用係数の台が有限である臨界コンタクトプロセスを考え,時空間n点関数のスケーリング極限が超ブラウン運動標準測度のn-1多点関数に一致することを証明した.

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