Research Highlight: Field theory for loop-erased random walks

The loop-erased random walk (LERW) was introduced by Lawler (1980) as a mathematically more tractable object than the self-avoiding walk (SAW). Nowadays, the LERW has numerous applications in combinatorics, self-organized criticality, conformal field theory and SLE. The SAW is defined as the uniform measure on all non-self-intersecting walks, and is well treatable by field-theoretic methods. The LERW is defined as the trajectory of a random walk in which any loop is erased as soon as it is formed:

A loop-erased random walk with its shadow,
courtesy of Michel Bauer and Denis Bernard

LERW plot

The number of steps t (not counting the erased loops) it takes to reach the distance L scales as

t ∼ L^d_f
where d_f is the fractal dimension of LERW. Contrary to the SAW, which, as shown by P.-G. de Gennes, is described by the O(N) model at N = 0, up to now there was no field-theoretic approach to compute d_f in a dimensional expansion around the upper critical dimension d_uc = 4. We proposed a field-theoretic description for the LERW, based on the Functional Renormalization Group (FRG), a method developed to study disordered systems. We build on a long chain of mappings: depinning transition of a periodic elastic system in random media ⇒ sandpile models ⇒ uniform spanning trees ⇒ LERW. To second order in ε = 4 – d we obtain

This is consistent with the exact value d_f =5/4 derived using conformal field theory in d = 2. In d = 3 the Padé approximation gives d_f = 1.614 ± 0.011, in fairly good agreement with the numerical estimation d_f = 1.6183 ± 0.0004. The FRG passes all the tests of presently known results for LERW. In particular, it reproduces the correct leading logarithmic corrections at the upper critical dimension and makes a prediction for the subleading logarithmic correction:
log correction to LERW

which remains to be tested in numerical simulations.

A.A. Fedorenko, P. Le Doussal, and K.J. Wiese, arXiv:0803.2357