arXiv subjects
Publications and source records attributed to Ellen Powell.
We study how the Gaussian multiplicative chaos (GMC) measures $μ^γ$ corresponding to the 2D Gaussian free field change when $γ$ approaches the critical parameter $2$. In particular, we show that as $γ\to 2^{-}$, $(2-γ)^{-1}μ^γ$ converges in probability to $2μ'$, where $μ'$ is the critical GMC measure.
We study branching diffusions in a bounded domain $D$ of $\mathbb{R}^d$ in which particles are killed upon hitting the boundary $\partial D$. It is known that any such process undergoes a phase transition when the branching rate $β$ exceeds a critical value: a multiple of the first eigenvalue of the generator of the diffusion. We investigate the system at criticality and show that the associated genealogical tree, when the process is conditioned to survive for a long time, converges to Aldous' Continuum Random Tree under appropriate rescaling. The result holds under only a mild assumption on the domain, and is valid for all branching mechanisms with finite variance, and a general class of diffusions.
We provide new constructions of the subcritical and critical Gaussian multiplicative chaos (GMC) measures corresponding to the 2D Gaussian free field (GFF). As a special case we recover E. Aidekon's construction of random measures using nested conformally invariant loop ensembles, and thereby prove his conjecture that certain CLE$_4$ based limiting measures are equal in law to the GMC measures for the GFF. The constructions are based on the theory of local sets of the GFF and build a strong link between multiplicative cascades and GMC measures. This link allows us to directly adapt techniques used for multiplicative cascades to the study of GMC measures of the GFF. As a proof of principle we do this for the so-called Seneta--Heyde rescaling of the critical GMC measure.
We study the level lines of a Gaussian free field in a planar domain with general boundary data $F$. We show that the level lines exist as continuous curves under the assumption that $F$ is regulated (i.e., admits left and right limits at every point), and satisfies certain inequalities. Moreover, these level lines are a.s. determined by the field. This allows us to define and study a generalization of the SLE$_4(\underlineρ)$ process, now with a continuum of force points. A crucial ingredient is a monotonicity property in terms of the boundary data which strengthens a result of Miller and Sheffield and is also of independent interest.