Cambridge Equinox
Probability Conference

The conference will focus on recent developments in probability theory. It will take place at the Centre for Mathematical Sciences.

Mini-Courses

• Marek Biskup (UCLA)
  
  A renormalization-based approach to the hierarchical integer-valued Gaussian free field

  The integer-valued Gaussian free field can be thought of as the continuum-valued Gaussian free field conditioned on taking values in integer multiples of the root-of-the-inverse-temperature. For the setting where the continuum field exhibits logarithmic correlations, the conditioning leads to a BKT-phase transition: the discrete-valued field remains rough, continuum-valued like, at high temperatures but is smooth, with toned-down correlations, at low temperatures. I will describe a renormalization-group based approach to this transition in the context of the hierarchical model (which is a priori log correlated) where this approach is relatively easy to implement. While particular instances of such analysis are classical, dating back to mid 1980s, recent innovations (obtained jointly with Haiyu Huang) permit seamless extension all the way up to, including and even going somewhat beyond the critical point.
• Grégory Miermont (ENS de Lyon)
  
  Scaling limits of random maps with large faces

  Random maps, which are simply random gluings of Euclidean polygons, provide natural models of discrete random surfaces that are expected to converge to universal continuum models of 2-dimensional random geometry, given in particular by Liouville quantum gravity surfaces. This is particularly well-understood in the context of "Brownian geometry" —a.k.a. LQG(\(\sqrt{8/3}\)) surfaces— which corresponds to limits of random maps with tame face degrees and in which the gluing is uniformly chosen among all possibilities. In this mini-course, we will study the situation where the gluing is still uniformly random, but where the faces are allowed to obey power-law degree distributions, leading in the scaling limit to random Sierpinski carpet or gasket-like objects. In the spirit of Pitman's "combinatorial stochastic processes", we will give a description of these limiting spaces in terms of classical objects of probability theory, namely, Lévy processes, and Poisson random measures, and discuss its relationship with conformal loop ensembles. This is based on joint work with Nicolas Curien and Armand Riera.

Program

Monday, 21 September

Tuesday, 22 September

Wednesday, 23 September

Thursday, 24 September

Friday, 25 September