In this talk, we focus on the second part of the proof: the conditional distribution of GUE's spectrum. The main ideas are the uniform distribution for eigenvalues of Hermitian matrices conditional on an isotropic hyperplane, and the SU-invariance of GUE. Moreover, I will give some combinatoric results for LPP model under i.i.d. Bernoulli environment.

In this talk, we focus on the first part of the proof: the combinatoric structure of LPP model. The essential key depends on the specific random environment of i.i.d. geometric distribution, which can be identified by the matrices with non-negative integer entries, and the RSK correspondence yields some exactly solvable formulas for LPP model.

In 2001, Yuliy Baryshnikov [1] connected the limiting process of last passage percolation (LPP) model under any i.i.d. non-negative random environment and the eigenvalues of Gaussian unitary ensemble (GUE) minors. The proof can be divided to two parts: the combinatoric structure of LPP model and the conditional distribution of GUE's spectrum. In this talk, we introduce Baryshnikov's result and the combinatoric tool, Robinson-Schensted-Knuth (RSK) correspondence related to the LPP model.

[1] Y Baryshnikov. GUEs and queues. Probab. Theory Related Fields, 2001.

In this talk, we unite the previously discussed concepts to finish the proof of the Parisi Formula.

In this talk, we begin with Guerra's Replica Symmetry Breaking Interpolation, deriving the Parisi Formula as an upper bound on free energy in the thermodynamic limit. We then introduce the Aizenman-Sims-Starr scheme, which can provide matching lower bounds on free energy.

We will proving the Ghirlanda-Guerra identities and the Talagrand positivity principle is this talk, following Chapter 3 of Panchenko's book.

We will start proving the Parisi formula for mean field spin glasses, following Chapter 3 of Panchenko's book. Knowledge on the previous chapters is not necessary but will be helpful.