Math 583 A

Random Matrix Techniques For Sparse Graphs

Spring 2013

• Meeting:                    MWF, 1:30pm-2:20 pm, PDL C-401
  
• Office hours:              W 2:30-5pm; also by appointment
  

• Instructors:               Ioana Dumitriu (dumitriu at math)

                                   Soumik Pal (soumik at math)

• Bibliography

• For more on weak graph limits, see Justin Salez's PhD thesis (chapter 2 in particular)
• For universality of random matrices, see the following series of papers by Tao and Vu:
• Random matrices: Universality of local eigenvalue statistics (2010)
• Random matrices: Universality of local eigenvalue statistics up to the edge (2010)
• Random matrices: The Four Moment Theorem for Wigner Ensembles (2011)
• This paper (Dumitriu-Pal) shows the local semicircle law and eigenvector delocalization for d-regular graphs. 
• For a different proof (in a slightly different regime) of the local semicircle law for Erdos-Renyi and d-regular graphs, see  this paper by Linh Tran, Van Vu, and Ke Wang:
  
• Sparse Random Graphs: Eigenvalues and Eigenvectors (2010)
• Global fluctuations for the random graph can be found here.
• And here you can find the Gaussian Free Field process for the same model.

• List of Papers / Resources

• SYLVESTER, May 24: Trotter,  Eigenvalue Distributions of Large Hermitian Matrices; Wigner’s Semi-circle Law and a Theorem of Kac, Murdock, and Szego

• Bryc, Dembo, and Jiang, Spectral Measure of Large Hankel, Markov and Toeplitz Matrices

• Jiang, Empirical Distributions of Laplacian Matrices of Large Dilute Random Graphs

• Brooks and Lindenstrass, Non-localization of Eigenfunctions on Large Regular Graphs

• GERANDY, June 3: Bordenave, Caputo, and Chafai, Circular Law Theorem for Random Markov Matrices

• Arratia and Tavare, The Cycle Structure of Random Permutations

• Diaconis and Evans, Linear Functionals of Eigenvalues of Random Matrices

• Ferrari and Frings, On the partial connection between random matrices and interacting particle systems

• ANH, May 29: Kim and Vu, Sandwiching random graphs: universality between random graph models

• O'Connell, Random matrices, non-colliding processes, and queues

• REZA, June 7: Bhamidi, Evans, and Sen, Spectra of Random Trees

• Warren, Dyson’s Brownian motions, intertwining and interlacing