LERW as an Example of Off-Critical SLEs

摘要

Two dimensional loop erased random walk (LERW) is a random curve, whose continuum limit is known to be a Schramm-Loewner evolution (SLE) with parameter κ=2. In this article we study “off-critical loop erased random walks”, loop erasures of random walks penalized by their number of steps. On one hand we are able to identify counterparts for some LERW observables in terms of symplectic fermions (c=−2), thus making further steps towards a field theoretic description of LERWs. On the other hand, we show that it is possible to understand the Loewner driving function of the continuum limit of off-critical LERWs, thus providing an example of application of SLE-like techniques to models near their critical point. Such a description is bound to be quite complicated because outside the critical point one has a finite correlation length and therefore no conformal invariance. However, the example here shows the question need not be intractable. We will present the results with emphasis on general features that can be expected to be true in other off-critical models.