We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise -that is, we modulate the noise by a factor ε ց 0 -and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times exp cε −2 , c > 0, and we show both that on the time scale ε −2 the dephasing (i.e., the difference between noiseless and noisy system measured in a natural coordinate system that involves a phase) is close to a Brownian motion with constant drift, and that on longer time scales the dephasing dynamics is dominated, to leading order, by the drift. The natural choice of coordinates, that reduces the dynamics in a neighborhood of the cycle to a rotation, plays a central role and makes the connection with the applied science literature in which noisy limit cycle dynamics are often reduced to a diffusion model for the phase of the limit cycle.2010 Mathematics Subject Classification: 60H10, 34F05, 60F17, 82C31, 92B25

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We give a simplified proof for the equivalence of loop-erased random walks to a lattice model containing two complex fermions, and one complex boson. This equivalence works on an arbitrary directed graph. Specifying to the dd-dimensional hypercubic lattice, at large scales this theory reduces to a scalar \phi^4ϕ4-type theory with two complex fermions, and one complex boson. While the path integral for the fermions is the Berezin integral, for the bosonic field we can either use a complex field \phi(x)\in \mathbb Cϕ(x)∈ℂ (standard formulation) or a nilpotent one satisfying \phi(x)^2 =0ϕ(x)2=0. We discuss basic properties of the latter formulation, which has distinct advantages in the lattice model.