Abstract
AbstractThe random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm–Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, “large” lattice loops from the random walk loop soup converge to “large” loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soup and CLE goes via cluster boundaries. In this paper, we study the scaling limit of clusters of “large” lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of “large” lattice loops converges to CLE.
Highlights
Several interesting models of statistical mechanics, such as percolation and the Ising and Potts models, can be described in terms of clusters
What makes Schramm–Loewner evolution (SLE) and conformal loop ensembles (CLEs) natural candidates is their conformal invariance, a property expected of the scaling limit of two-dimensional statistical mechanical models at the critical point
For 8/3 < κ ≤ 4, CLEκ can be obtained [25] from the Brownian loop soup, introduced by Lawler and Werner [18], as we explain below
Summary
Several interesting models of statistical mechanics, such as percolation and the Ising and Potts models, can be described in terms of clusters. The continuum scaling limit of the collection of all macroscopic outer boundaries of critical Ising spin clusters is conjectured to correspond to CLE3 and to a Brownian loop soup with λ = 1/4. The connection between the Brownian loop soup and SLE/CLE goes through its loop clusters and their boundaries In view of this observation, it is interesting to investigate whether the random walk loop soup converges to the Brownian loop soup in terms of loop clusters and their boundaries, not just in terms of individual loops, as established by Lawler and Trujillo Ferreras [17]. The result announced appears to use our convergence result in a crucial way, combined with a coupling between the random walk loop soup and the Gaussian free field, and would give the convergence of the random walk loop soup to the Brownian loop soup keeping all loops
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
