ISSN:
1432-2064
Keywords:
82B41
;
60K35
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary We consider simple random walk onZ d perturbed by a factor exp[βT −P J T], whereT is the length of the walk and $$J_T = \sum\nolimits_{0 \leqslant i〈 j \leqslant T} \delta _{\omega (i),\omega (j)} $$ . Forp=1 and dimensionsd≥2, we prove that this walk behaves diffusively for all − ∞ 〈 β 〈0, with β0 〉 0. Ford〉2 the diffusion constant is equal to 1, but ford=2 it is renormalized. Ford=1 andp=3/2, we prove diffusion for all real β (positive or negative). Ford〉2 the scaling limit is Brownian motion, but ford≤2 it is the Edwards model (with the “wrong” sign of the coupling when β〉0) which governs the limiting behaviour; the latter arises since for $$p = \frac{{4 - d}}{2}$$ ,T −p J T is the discrete self-intersection local time. This establishes existence of a diffusive phase for this model. Existence of a collapsed phase for a very closely related model has been proven in work of Bolthausen and Schmock.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01195476
Permalink
