18 research outputs found
Extinction threshold in the spatial stochastic logistic model: space homogeneous case
We consider the extinction regime in the spatial stochastic logistic model in R^d (a.k.a. Bolker–Pacala–Dieckmann–Law model of spatial populations) using the first-order perturbation beyond the mean-field equation. In space homogeneous case (i.e. when the density is non-spatial and the covariance is translation invariant), we show that the perturbation converges as time tends to infinity; that yields the first-order approximation for the stationary density. Next, we study the critical mortality – the smallest constant death rate which ensures the extinction of the population – as a function of the mean-field scaling parameter ε>0. We find the leading term of the asymptotic expansion (as ε→0) of the critical mortality which is apparently different for the cases d≥3, d = 2, and d = 1
Nonlinear Sigma Model for Disordered Media: Replica Trick for Non-Perturbative Results and Interactions
In these lectures, given at the NATO ASI at Windsor (2001), applications of
the replicas nonlinear sigma model to disordered systems are reviewed. A
particular attention is given to two sets of issues. First, obtaining
non-perturbative results in the replica limit is discussed, using as examples
(i) an oscillatory behaviour of the two-level correlation function and (ii)
long-tail asymptotes of different mesoscopic distributions. Second, a new
variant of the sigma model for interacting electrons in disordered normal and
superconducting systems is presented, with demonstrating how to reduce it,
under certain controlled approximations, to known ``phase-only'' actions,
including that of the ``dirty bosons'' model.Comment: 25 pages, Proceedings of the NATO ASI "Field Theory of Strongly
Correlated Fermions and Bosons in Low - Dimensional Disordered Systems",
Windsor, August, 2001; to be published by Kluwe