Random Triangulations Coupled with Ising Model

par Laurent Ménard (Paris Nanterre)

vendredi 2 févr. 2024, 14:00 → 16:00 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Probability and analysis informal seminar

In this talk, we will investigate geometric properties of random planar triangulations coupled with an Ising model. This model is known to undergo a combinatorial phase transition at an explicit critical temperature, for which its partition function has a different asymptotic behavior than uniform maps. I will briefly explain this phenomenon, and why it hints at a different universality class than the Brownian sphere.
In the second part of the talk, we will focus on the geometry of spin clusters in the infinite volume setting. We will exhibit a phase transition for the existence of an infinite spin cluster: for critical and supercritical temperatures, the root spin cluster is finite almost surely, while it is infinite with positive probability for subcritical temperatures. A lot of precise information can be derived in all regimes. In particular, we will see that in the whole supercritical temperature regime, critical exponents for spin clusters are the same as for critical Bernoulli site percolation on uniform planar triangulations.
Based on joint works with Marie Albenque and Gilles Schaeffer.

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