Field theories for loop-erased random walks

Field theory conjecture for loop - erased random walks

We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the loop-erased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in ε = 4− d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critic...

Generalized loop-erased random walks and approximate reachability

In this paper we extend the loop-erased random walk (LERW) to the directed hypergraph setting. We then generalize Wilson’s algorithm for uniform sampling of spanning trees to directed hypergraphs. In several special cases, this algorithm perfectly samples spanning hypertrees in expected polynomial time. Our main application is to the reachability problem, also known as the directed all-terminal...

Loop-erased Random Walks, Spanning Trees and Hamiltonian Cycles

We establish a formula for the distribution of loop-erased random walks at certain random times. Several classical results on spanning trees, including Wilson’s algorithm, follow easily, as well as a method to construct random Hamiltonian cycles.

Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions.

We show that in the loop-erased random-walk problem, the exponent characterizing the probability distribution of areas of erased loops is superuniversal. In d dimensions, the probability that the erased loop has an area A varies as A(-2) for large A, independent of d, for 2< or =d< or =4. We estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d=2 an...

Loop-Erased Random Walk