Random Concave Functions on an Equilateral Lattice with Periodic Hessians

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Characterizes all concave functions on the equilateral triangular lattice whose second differences (Hessians) repeat periodically, proves existence and uniqueness of an ergodic Gibbs measure conditioned on a periodic Hessian, and shows spatial averages satisfy a CLT with variance tending to a nonzero constant. Method uses explicit construction of periodic discrete potentials, discrete Fourier diagonalization of the periodic difference operator, and entropy/spectral-gap estimates to control fluctuations. Provides numerics on finite tori that match the predicted fluctuation amplitude and verifies the discrete elliptic inversion used in proofs.

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" chapters (20)

1. 0:00Intro
2. 0:26Connection to representation theory
3. 5:19Connection to statistical physics
4. 6:00Littlewood-Richardson coefficients and mosaics
5. 7:41Connection to random matrices
6. 8:30Scaling limits
7. 9:33Discrete Hessian
8. 10:17Average Hessian
9. 10:57Periodic Hessians
10. 16:14Volume of P,(s)
11. 24:22Prekopa-Leindler
12. 25:36Scheme of proof of Theorems 1 and 2
13. 27:01Details: studying fluctuations using characters
14. 27:33Reduction to complex
15. 28:42exponentials: idea behind proof
16. 30:16An upper bound on the local volumes.
17. 30:43Boundary vertices
18. 39:14Theorem 2
19. 41:26Dirichlet boundary conditions
20. 41:57Open problem: Exact computation of entropy