A generalised inductive approach to the lace expansion

We study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions $d=1,2$, where $x$ and $y$ a re connected with probability $\sim\beta/\|x-y\|^{-s}$. We show that when $d<s<2d$ the walk is transient, and when $s\geq 2d$, the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension $d$, if $d<s<2d$, then critical percola...

Let $p_c(\mathbb{Q}_n)$ and $p_c(\mathbb{Z}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$-cube $\mathbb{Q}_n = \{0,1\}^n$ and on $\Z^n$, respectively. Let $\Omega = n$ for $\mathbb{G} = \mathbb{Q}_n$ and $\Omega = 2n$ for $\mathbb{G} = \mathbb{Z}^n$ denote the degree of $\mathbb{G}$. We use the lace expansion to prove that for both $\mathbb{G} = \mathbb{Q}_n$ and $\mathbb{G} = \mathbb{Z}^n$, $p_c(\mathbb{G}) & = \cn^{-1} + \cn^{-2} + {7/2}...

We consider the spread-out models of the self-avoiding walk and its finite-memory version, called the memory-$\tau$ walk. For both models, each step is uniformly distributed over the d-dimensional box $\{x\in\mathbb Z^d:\|x\|_{\infty} \le L\}$. The critical point $p_c^\tau$ for the memory-$\tau$ walk is increasing in $\tau$ and converges to the critical point $p_c^\infty$ for the self-avoiding walk as $\tau\uparrow\infty$. The best estimate of the rate of convergence so far w...

A spread-out lattice animal is a finite connected set of edges in $\{ \{x,y\} \subset \mathbb{Z}^d:0<||x-y||\le L \}$. A lattice tree is a lattice animal with no loops.The best estimate on the critical point $p_c$ so far was achieved by Penrose(JSP,77(1994):3-15): $p_c=1/e+O(L^{-2d/7}\log L)$ for both models for all $d\ge1$. In this paper, we show that $p_c=1/e+CL^{-d}+O(L^{-d-1})$ for all $d>8$, where the model-dependent constant $C$ has the random-walk representation $C_\ma...

We consider percolation on $\mathbb{Z}^d$ and on the $d$-dimensional discrete torus, in dimensions $d \ge 11$ for the nearest-neighbour model and in dimensions $d>6$ for spread-out models. For $\mathbb{Z}^d$, we employ a wide range of techniques and previous results to prove that there exist positive constants $c$ and $C$ such that the slightly subcritical two-point function and one-arm probabilities satisfy \[ \mathbb{P}_{p_c-\varepsilon}(0 \leftrightarrow x) \leq \frac{C}{\...

These lecture notes provide a rapid introduction to a number of rigorous results on self-avoiding walks, with emphasis on the critical behaviour. Following an introductory overview of the central problems, an account is given of the Hammersley--Welsh bound on the number of self-avoiding walks and its consequences for the growth rates of bridges and self-avoiding polygons. A detailed proof that the connective constant on the hexagonal lattice equals $\sqrt{2+\sqrt{2}}$ is then...

We consider the cardinality of supercritical oriented bond percolation in two dimensions. We show that, whenever the origin is conditioned to percolate, the process appropriately normalized converges asymptotically in distribution to the standard normal law. This resolves a longstanding open problem pointed out to in several instances in the literature. The result applies also to the continuous-time analog of the process, viz. the basic one-dimensional contact process. We als...

Let $X_1, X_2, \ldots$ be i.i.d. random variables with values in $\mathbb{Z}^d$ satisfying $\mathbb{P} \left(X_1=x\right) = \mathbb{P} \left(X_1=-x\right) = \Theta \left(\|x\|^{-s}\right)$ for some $s>d$. We show that the random walk defined by $S_n = \sum_{k=1}^{n} X_k$ is recurrent for $d\in \{1,2\}$ and $s \geq 2d$, and transient otherwise. This also shows that for an electric network in dimension $d\in \{1,2\}$ the condition $c_{\{x,y\}} \leq C \|x-y\|^{-2d}$ implies recu...

We study biased random walk on the infinite connected component of supercritical percolation on the integer lattice $\mathbb{Z}^d$ for $d\geq 2$. For this model, Fribergh and Hammond showed the existence of an exponent $\gamma$ such that: for $\gamma<1$, the random walk is sub-ballistic (i.e. has zero velocity asymptotically), with polynomial escape rate described by $\gamma$; whereas for $\gamma>1$, the random walk is ballistic, with non-zero speed in the direction of the bi...

We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\geq 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show that the level set of the zero-average Gaussian free field above level $h$ exhibits a phase transition at level $h_\star$, which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite $d$-regular t...