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The goal of this book is to present new mathematical techniques for studyingthe behaviour of mean-field systems with disordered interactions. We mostlyfocus on certain problems of statistical inference in high dimension, and onspin glasses. The techniques we present aim to determine the free energy ofthese systems, in the limit of large system size, by showing that theyasymptotically satisfy a Hamilton-Jacobi equation.  The first chapter is a general introduction to statistical mechanics, with afocus on the Curie-Weiss model. We give a brief introduction to convex analysisand large deviation principles in Chapter 2, and identify the limit free energyof the Curie-Weiss model using these tools. In Chapter 3, we define the notionof viscosity solution to a Hamilton-Jacobi equation, and use it to recover thelimit free energy of the Curie-Weiss model. We discover technical challenges toapplying the same method to generalized versions of the Curie-Weiss model, anddevelop a new selection principle based on convexity to overcome these. We thenturn to statistical inference in Chapter 4, focusing on the problem ofrecovering a large symmetric rank-one matrix from a noisy observation, and wesee that the tools developed in the previous chapter apply to this setting aswell. Chapter 5 is preparatory work for a discussion of the more challengingcase of spin glasses. The first half of this chapter is a self-containedintroduction to Poisson point processes, including limit theorems on extremevalues of independent and identically distributed random variables, which webelieve to be of independent interest. We finally turn to the setting of spinglasses in Chapter 6. For the Sherrington-Kirkpatrick model, we show how torelate the Parisi formula with the Hamilton-Jacobi approach. We conclude with amore informal discussion on the status of current research for more challengingmodels.

Statistical mechanics of mean-field disordered systems: a Hamilton-Jacobi approach