In this talk, I will present recent results, obtained in collaboration with Laurent Ménard, about the geometry of spin clusters in Ising-decorated triangulations, and build on previously work obtained in collaboration with Laurent Ménard and Gilles Schaeffer.
In this model, triangulations are sampled together with a spin configuration on their vertices, with a probability biased by their number of monochromatic edges, via a parameter nu. The fact that there exists a combinatorial critical value for this model has been initially established in the physics literature by Kazakov and was rederived by combinatorial methods by Bousquet-Mélou and Schaeffer, and Bouttier, Di Francesco and Guitter.
Here, we give geometric evidence of that this model undergoes a phase transition by studying the volume and perimeter of its monochromatic clusters. In particular, we establish that, when nu is critical or subcritical, the cluster of the root is finite almost surely, and is infinite with positive probability for nu supercritical.[-]

I will first introduce a general class of mean-field-like spin systems with random couplings that comprises both the Ising model on inhomogeneous dense random graphs and the randomly diluted Hopfield model. I will then present quantitative estimates of metastability in large volumes at fixed temperatures when these systems evolve according to a Glauber dynamics, i.e. where spins flip with Metropolis rates at inverse temperature $\beta $. The main result identifies conditions ensuring that with high probability the system behaves like the corresponding system where the random couplings are replaced by their averages. More precisely, we prove that the metastability of the former system is implied with high probability by the metastability of the latter. Moreover, we consider relevant metastable hitting times of the two systems and find the asymptotic tail behaviour and the moments of their ratio. This result provides an extension of the results known for the Ising model on the the Erdos-Renyi random graph. Our proofs use the potential-theoretic approach to metastability in combination with concentration inequalities.
Based on a joint work in collaboration with Anton Bovier, Frank den Hollander, Saeda Marello and Martin Slowik.[-]

The Gibbs measure of many disordered systems at low temperature may exhibit a very strong dependance on even tiny variations of temperature, usually called “temperature chaos”. I will discuss this question for Spin Glasses. I will report on a recent work with Eliran Subag (Courant) and Ofer Zeitouni (Weizmann and Courant), where we give a detailed geometric description of the Gibbs measure at low temperature, which in particular implies temperature chaos for a general class of spherical Spin Glasses at low temperature. This question has a very long past in the physics literature, and an interesting recent history in mathematics. Indeed, in 2015, Eliran Subag has given a very sharp description of the Gibbs measure for pure p-spin spherical Spin Glasses at low temperature, building on results on the complexity of these spin glasses by Auffinger-Cerny and myself. This description (close to the so-called Thouless-Anderson-Palmer picture) excludes the existence of temperature chaos for the pure p-spin!! The recent work gives an extension of this very detailed geometric description of the Gibbs measure to the case of general mixed models, and shows that in fact the pure p-spin is very singular.[-]