UC Berkeley Discrete Analysis Seminar, Spring 2026

In this talk, we will consider the recent result of Mohanty and Rajaraman giving new bounds for concentration and expectations of spectral radii for various families of random matrices. Their technique proceeds by application of Jensen's formula to the characteristic polynomial. In this first talk, we will focus on establishing their main general results and the application to Girko matrices. Subsequently we will discuss the machinery for extending these results to Wigner matrices and random regular graphs via a more general condition.

We continue with the recent paper of Mohanty and Rajaraman by discussing their general condition on families of random matrices which implies a spectral radius bound. The first part of the talk will cover the application of the condition to yield the bound; the second part will focus on verifying the condition for adjacency matrices of random regular graphs.

We consider a recent paper by Hsieh, Mohanty and Zhang which gives a simple construction of almost-optimal balanced codes. Previous constructions relied on analyzing codes constructed using random walks in an expander graph; the new insight is to use random walks where each step switches to a new expander graph. The proof rests on bounding the spectral norm of a product involving the adjacency matrices of each expander. The first talk will introduce codes and how expanders can be used to construct them, as well as the achievements of this new construction.

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