Critical and Near-critical Interfaces

The reading group will focus on random interfaces that should arise as the scaling limits of massive/near-critical models in statistical mechanics. The most well-known critical interface models are the Schramm-Loewner Evolution (SLE) and Conformal Loop Ensembles (CLE). While the study of critical interfaces has seen many advances in the past two decades, a general theory for near-critical interfaces remains elusive. However, a few rigorous examples of massive interface models have been constructed in the last few years, which we will also explore.

We will begin by discussing the general properties of SLE and providing an overview of the expected properties of massive interfaces. From there, we will explore other important constructions for critical interfaces, followed by descriptions of a few examples of massive interfaces that have already been described. Finally, we will review some results in the physics literature that are related to massive models.

Contributing to the reading group

All members of Durham University are welcome to join the group. Presenting is highly encouraged, but not mandatory. However, if you wish to contribute to the reading group in different way, you can send me your notes (handwritten or latex) from the talks and I can put them here.

Time and place

We are planning to meet weekly on Thursdays at 15h00 for 60’ to 90’ and have 9 or 10 meetings in total. Please let me know if you are interested but cannot join at this time. We will meet at the magic room.

Zoom

We will also transmit the meetings via zoom for those interested. However, we note that we do not guarantee ease of following on zoom. Furthermore, speakers are not expected to write slides or anything of the sort.

Zoom Link Passcode will be provided via email.

(Provisional) Schedule

  • 11/01/2024: Ellen Powell - Intro to SLE (see extra material 1)
  • 18/01/2024: Leandro Chiarini - [6]
  • 25/01/2024: Ellen Powell - [4]
  • 01/02/2024: Léonie Papon - [5]
  • 08/02/2024: Léonie Papon - Dynkin’s isomorphism for the GFF CLE couplings for massive and massless cases [3]
  • 15/02/2024: Leandro Chiarini - [2]
  • 22/02/2024: To confirm - [1]
  • 29/02/2024: To confirm - [1]
  • 07/03/2024: To confirm - To confirm
  • 14/03/2024: To confirm - To confirm

Notes from talks

References (More to be added)

  1. Berestycki, Haunschmid-Sibitz, “Near-critical dimers and massive SLE”. Link
  2. Camia, “Off-criticality and the massive Brownian loop soup”. Link
  3. Le Jan, “Markov paths, loops and fields”. Link
  4. Makarov, Smirnov “Off-critical lattice models and massive SLEs”. Link
  5. Papon,“Massive SLE4, massive CLE4 and the massive planar GFF”. Link
  6. Sheffield, Werner, “Conformal loop ensembles: the Markovian characterization and the loop-soup construction”. Link
    • Image above from Wikipedia

Extra material

  1. Berestycki, Powell, “Gaussian free field and Liouville quantum gravity” Link
  2. Kemppainen, “Schramm–Loewner evolution”

Organised by