Monotonicity for excited random walk in high dimensions
We prove that the drift $θ(d,β)$ for excited random walk in dimension $d$ is monotone in the excitement parameter $β\in[0, 1]$, when $d\ge 9$.
math.PR↗
arXiv subjects
Mark Holmes
Publications and source records attributed to Mark Holmes.
Weak convergence of measure-valued processes and $r$-point functions
We prove a sufficient set of conditions for a sequence of finite measures on the space of cadlag measure-valued paths to converge to the canonical measure of super-Brownian motion in the sense of convergence of finite-dimensional distributions. The conditions are convergence of the Fourier transform of the $r$-point functions and perhaps convergence of the ``survival probabilities.'' These conditions have recently been shown to hold for a variety of statistical mechanical models, including critical oriented percolation, the critical contact process and lattice trees at criticality, all above their respective critical dimensions.
math.PR↗
An extension of the inductive approach to the lace expansion
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 asymptotic behaviour for the Fourier transform of the two-point function for sufficiently spread-out lattice trees in dimensions d>8, and it is potentially also applicable to percolation in dimensions d>6.
math.PR↗
Extension of the generalised inductive approach to the lace expansion: Full proof
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 sufficiently spread-out lattice trees in dimensions d>8 and may also be applicable to percolation in dimensions d>6.
math.PR↗