Understanding the (near-)critical behaviour of lattice models is one of the main challenges in statistical mechanics. A prominent approach to this problem is the computation of the model’s critical exponents. This task is generally impossible due to the intricate interplay between the specific features of the models and the geometry of the graphs on which they are defined. A striking observation was made in the case of models defined on Z^d: beyond an upper-critical dimension d_c, the geometry no longer plays a significant role, and the critical exponents simplify, matching those found on Cayley trees or complete graphs. The regime d>d_c is called the mean-field regime of the model.

In the 1980’s, two prominent approaches have been developed to understand the mean-field regime of a wide class of models: the rigorous renormalization group method and the lace expansion. It usually requires a lot of work to transfer the analysis involved in these methods from one model to the other. 

We revisit the study of the mean-field regime and present an alternative, more probabilistic, and unified approach. Our method applies to many perturbative settings including, the weakly-self avoiding walk model in dimensions d>4, spread-out Bernoulli Percolation in dimensions d>6, or even one- and two-component spin models in dimensions d>4. 

Based on ongoing works with Hugo Duminil-Copin, Aman Markar, and Gordon Slade.