From 1 to 6: A Finer Analysis of Perturbed Branching Brownian Motion

Abstract

We derive a multiscale generalisation of the Bakry-Émery criterion for a measure to satisfy a log-Sobolev inequality. Our criterion relies on the control of an associated PDE well-known in renormalisation theory: the Polchinski equation. It implies the usual Bakry-Émery criterion, but we show that it remains effective for measures that are far from log-concave. Indeed, using our criterion, we prove that the massive continuum sine-Gordon model with < 6 satisfies asymptotically optimal log-Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory.

Anomalous spreading in reducible multitype branching Brownian motion

“…Understanding the behaviour of the process at these points could help understanding the phase transitions occurring between the different areas of the state space. This would allow results similar to the ones developed in [14] for time-inhomogeneous BBM to be considered in reducible multitype BBM.…”

Extremes of the 2d scale-inhomogeneous discrete Gaussian free field: Extremal process in the weakly correlated regime

“…In Aïdékon et al (2013); Arguin et al (2013), the cluster process was shown to be BBM conditioned on the maximum being larger than √ 2t, or alternatively given as the limiting distribution of the neighbours of a local maximum. The extremal process of variable-speed BBM was investigated in Bovier andHartung (2014, 2015); Maillard and Zeitouni (2016); Fang and Zeitouni (2012); Bovier and Hartung (2020). In the regime of weak correlations, i.e.…”