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June, 2020 A Survey on the Lace Expansion for the Nearest-neighbor Models on the BCC Lattice
Satoshi Handa, Yoshinori Kamijima, Akira Sakai
Taiwanese J. Math. 24(3): 723-784 (June, 2020). DOI: 10.11650/tjm/190904


The aim of this survey is to explain, in a self-contained and relatively beginner-friendly manner, the lace expansion for the nearest-neighbor models of self-avoiding walk and percolation that converges in all dimensions above 6 and 9, respectively. To achieve this, we consider a $d$-dimensional version of the body-centered cubic (BCC) lattice, on which it is extremely easy to enumerate various random-walk quantities. Also, we choose a particular set of bootstrapping functions, by which a notoriously complicated part of the lace-expansion analysis becomes rather transparent.


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Satoshi Handa. Yoshinori Kamijima. Akira Sakai. "A Survey on the Lace Expansion for the Nearest-neighbor Models on the BCC Lattice." Taiwanese J. Math. 24 (3) 723 - 784, June, 2020.


Received: 17 May 2019; Accepted: 25 September 2019; Published: June, 2020
First available in Project Euclid: 19 May 2020

zbMATH: 07251195
MathSciNet: MR4100717
Digital Object Identifier: 10.11650/tjm/190904

Primary: 60K35 , 82B20 , 82B27 , 82B41 , 82B43

Keywords: Lace expansion , Mean-field behavior , percolation , Self-avoiding walk , upper critical dimension

Rights: Copyright © 2020 The Mathematical Society of the Republic of China


Vol.24 • No. 3 • June, 2020
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