## Taiwanese Journal of Mathematics

- Taiwanese J. Math.
- Advance publication (2019), 62 pages.

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

Satoshi Handa, Yoshinori Kamijima, and Akira Sakai

#### 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., Advance publication (2019), 62 pages.

**Dates**

First available in Project Euclid: 18 October 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.twjm/1571364135

**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., advance publication, 18 October 2019. doi:10.11650/tjm/190904. https://projecteuclid.org/euclid.twjm/1571364135