Related papers: The continuous-time lace expansion

The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we…

We show Green's function asymptotic upper bound for the two-point function of weakly self-avoiding walk in dimension bigger than 4, revisiting a classic problem. Our proof relies on Banach algebras to analyse the lace-expansion fixed point…

The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by…

We consider long-range percolation, Ising model, and self-avoiding walk on $\mathbb{Z}^d$, with couplings decaying like $|x|^{-(d+\alpha)}$ where $0 < \alpha \le 2$, above the upper critical dimensions. In the spread-out setting where the…

Using the Griffiths-Simon construction of the $\varphi^4$ model and the lace expansion for the Ising model, we prove that, if the strength $\lambda\ge0$ of nonlinearity is sufficiently small for a large class of short-range models in…

We derive a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the…

We extend the inductive approach to the lace expansion, previously developed to study models with critical dimension 4, to be applicable more generally. In particular, the result of this note has recently been used to prove Gaussian…

We study the random connection model driven by a stationary Poisson process. In the first part of the paper, we derive a lace expansion with remainder term in the continuum and bound the coefficients using a new version of the BK…

We use the lace expansion to give a simple proof that the critical two-point function for weakly self-avoiding walk on $\mathbb{Z}^d$ has decay $|x|^{-(d-2)}$ in dimensions $d>4$. The proof uses elementary Fourier analysis and the…

This paper extends the inductive approach to the lace expansion of van der Hofstad and Slade in order to prove Gaussian asymptotic behaviour for models with critical dimension other than 4. The results are applied by Holmes to study…

We extend and apply a rigorous renormalisation group method to study critical correlation functions, on the 4-dimensional lattice $\mathbb{Z}^4$, for the weakly coupled $n$-component $|\varphi|^4$ spin model for all $n \geq 1$, and for the…

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the critical point, and prove an upper bound…

We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on the d-dimensional hyper cubic lattice having long finite-range connections, above their upper critical dimensions d=4…

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…

Random walks on expanders play a crucial role in Markov Chain Monte Carlo algorithms, derandomization, graph theory, and distributed computing. A desirable property is that they are rapidly mixing, which is equivalent to having a spectral…

We prove $|x|^{-2}$ decay of the critical two-point function for the continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$, in the upper critical dimension $d=4$. This is a statement that the critical exponent $\eta$ exists and is…

We give conditions on a real-valued function $F$ on $\mathbb{Z}^d$, for $d>2$, which ensure that the solution $G$ to the convolution equation $(F*G)(x) = \delta_{0,x}$ has Gaussian decay $|x|^{-(d-2)}$ for large $|x|$. Precursors of our…

We prove a scaling limit theorem for the simple random walk on critical lattice trees in $\mathbb{Z}^d$, for $d\geq 8$. The scaling limit is the Brownian motion on the Integrated Super-Brownian Excursion (BISE) which is the same one that we…

By introducing a new measure for the infinite Galton-Watson process and providing estimates for (discrete) Green's functions on trees, we establish the asymptotic behavior of the capacity of critical branching random walks: in high…

For a random walk $S_n$ on $\mathbb{R}^d$ we study the asymptotic behaviour of the associated centre of mass process $G_n = n^{-1} \sum_{i=1}^n S_i$. For lattice distributions we give conditions for a local limit theorem to hold. We prove…