Self-avoiding walks and connective constants

The connective constant $\mu(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $\mu$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive cubic graphs (where $\phi$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $\bullet$ We present strict inequalities for the connective constants $\mu(G)$ of transitive graphs $G$, as $G$ varies. $\bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. $\bullet$ We describe so-called graph height functions within an account of "bridges" for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. $\bullet$ A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. $\bullet$ Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. $\bullet$ The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with arXiv:1304.721

Self-attracting self-avoiding walk

This article is concerned with self-avoiding walks (SAW) on $\mathbb{Z}^{d}$ that are subject to a self-attraction. The attraction, which rewards instances of adjacent parallel edges, introduces difficulties that are not present in ordinary SAW. Ueltschi has shown how to overcome these difficulties for sufficiently regular infinite-range step distributions and weak self-attractions. This article considers the case of bounded step distributions. For weak self-attractions we show that the connective constant exists, and, in $d\geq 5$, carry out a lace expansion analysis to prove the mean-field behaviour of the critical two-point function, hereby addressing a problem posed by den Hollander

Diffusive and Super-Diffusive Limits for Random Walks and Diffusions with Long Memory

We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in ${\Bbb R}^d$ or ${\Bbb Z}^d$. The first class consists of random walks on ${\Bbb Z}^d$ in divergence-free random drift field, modelling the motion of a particle suspended in time-stationary incompressible turbulent flow. The second class consists of self-repelling random diffusions, where the diffusing particle is pushed by the negative gradient of its own occupation time measure towards regions less visited in the past. We establish normal diffusion (with square-root-of-time scaling and Gaussian limiting distribution) in three and more dimensions and typically anomalously fast diffusion in low dimensions (typically, one and two). Results are quoted from various papers published between 2012-2018, with some hints to the main ideas of the proofs. No technical details are presented here.Comment: ICM-2018 Probability Section tal

Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals

We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ${\mathbb{Z}}^d$. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to $x\in{\mathbb{Z}}^d$, the probability of a connection from the origin to $x$, and the generating functions for lattice trees or lattice animals containing the origin and $x$. Using the lace expansion, we prove that the two-point function at the critical point is asymptotic to $\mathit{const.}|x|^{2-d}$ as $|x|\to\infty$, for $d\geq 5$ for self-avoiding walk, for $d\geq19$ for percolation, and for sufficiently large $d$ for lattice trees and animals. These results are complementary to those of [Ann. Probab. 31 (2003) 349--408], where spread-out models were considered. In the course of the proof, we also provide a sufficient (and rather sharp if $d>4$) condition under which the two-point function of a random walk on ${{\mathbb{Z}}^d}$ is asymptotic to $\mathit{const.}|x|^{2-d}$ as $|x|\to\infty$.Comment: Published in at http://dx.doi.org/10.1214/009117907000000231 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Expansion in high dimension for the growth constants of lattice trees and lattice animals

We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion-exclusion.Comment: 38 pages, 8 figures. Added section 6 to obtain the first term in the expansion, making the present paper more self-contained with very little change to the structure of the original paper. Accepted for publication in Combinatorics Probability and Computin