Taiwanese Journal of Mathematics

A Survey on the Lace Expansion for the Nearest-neighbor Models on the BCC Lattice

Satoshi Handa, Yoshinori Kamijima, and Akira Sakai

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Abstract

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.

Article information

Source
Taiwanese J. Math., Volume 24, Number 3 (2020), 723-784.

Dates
Received: 17 May 2019
Accepted: 25 September 2019
First available in Project Euclid: 19 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1589875227

Digital Object Identifier
doi:10.11650/tjm/190904

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B27: Critical phenomena 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82B43: Percolation [See also 60K35]

Keywords
self-avoiding walk percolation mean-field behavior upper critical dimension lace expansion

Citation

Handa, Satoshi; Kamijima, Yoshinori; Sakai, Akira. A Survey on the Lace Expansion for the Nearest-neighbor Models on the BCC Lattice. Taiwanese J. Math. 24 (2020), no. 3, 723--784. doi:10.11650/tjm/190904. https://projecteuclid.org/euclid.twjm/1589875227


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