Schramm's formula for multiple loop-erased random walks

We revisit the computation of the discrete version of Schramm's formula for the loop-erased random walk derived by Kenyon. The explicit formula in terms of the Green function relies on the use of a complex connection on a graph, for which a line bundle Laplacian is defined. We give explicit results in the scaling limit for the upper half-plane, the cylinder and the Moebius strip. Schramm's formula is then extended to multiple loop-erased random walks.Comment: 59 pages, 19 figures. v2: reformulation of Section 2.3, minor correction

Multipoint correlators in the Abelian sandpile model

We revisit the calculation of height correlations in the two-dimensional Abelian sandpile model by taking advantage of a technique developed recently by Kenyon and Wilson. The formalism requires to equip the usual graph Laplacian, ubiquitous in the context of cycle-rooted spanning forests, with a complex connection. In the case at hand, the connection is constant and localized along a semi-infinite defect line (zipper). In the appropriate limit of a trivial connection, it allows one to count spanning forests whose components contain prescribed sites, which are of direct relevance for height correlations in the sandpile model. Using this technique, we first rederive known 1- and 2-site lattice correlators on the plane and upper half-plane, more efficiently than what has been done so far. We also compute explicitly the (new) next-to-leading order in the distances ($r^{-4}$ for 1-site on the upper half-plane, $r^{-6}$ for 2-site on the plane). We extend these results by computing new correlators involving one arbitrary height and a few heights 1 on the plane and upper half-plane, for the open and closed boundary conditions. We examine our lattice results from the conformal point of view, and confirm the full consistency with the specific features currently conjectured to be present in the associated logarithmic conformal field theory.Comment: 60 pages, 21 figures. v2: reformulation of the grove theorem, minor correction

Sandpile probabilities on triangular and hexagonal lattices

We consider the Abelian sandpile model on triangular and hexagonal lattices. We compute several height probabilities on the full plane and on half-planes, and discuss some properties of the universality of the model.Comment: 26 pages, 12 figures. v2 and v3: minor correction

Sector-improved residue subtraction: Improvements and Applications

We discuss two recent developments of the sector-improved residue subtraction scheme for handling real radiation at NNLO in QCD. We present a new phase space construction which minimizes the number phase space configurations for subtraction terms and we rederive the four-dimensional formulation of the scheme.Comment: Contribution to the proceedings of the Loops and Legs 2018 conference, St. Goar, German