Take a uniformly random graph with a fixed edge density e. Its triangle density will typically be about e^3, and we are interested in the large deviations behavior: what's the probability that the triangle density is about e^3 - delta? The general theory for this sort of problem was studied by Chatterjee-Varadhan and Dembo-Lubetzky, who showed that the solution can be written in terms of an optimization over certain integral kernels. This optimization is difficult to solve explicitly, but Kenyon, Radin, Ren and Sadun used numerics to come up with a fascinating and intricate set of conjectures regarding both the probabilities and the structures of the conditioned random graphs. We prove these conjectures in a small region of the parameter space.

Joint work with Charles Radin and Lorenzo Sadun

https://sites.google.com/view/paw-seminar

https://sites.google.com/view/paw-seminar/

Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this talk, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a non-trivial signature, i.e. two finite sequences of non-increasing natural numbers, and for n large enough, consider the irreducible representation of Un associated to this signature. We show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. This is a joint work with Benoît Collins.

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In the 80’s Dahlberg asked two questions regarding the `$L^p$ – solvability’ of elliptic equations with variable coefficients. Dahlberg’s first question was whether $L^p$ solvability was maintained under `Carleson-perturbations’ of the coefficients. This question was answered by Fefferman, Kenig and Pipher [FKP], where they also introduced new characterizations of $A_\infty$, reverse-Hölder and $A_p$ weights. These characterizations were used to create a counterexample to show their theorem was sharp.

Dahlberg’s second question was whether a Carleson gradient/oscillation condition (the `DKP condition’) was enough to imply $L^p$ solvability for some p > 1. This was answered by Kenig and Pipher [KP] and refined by Dindos, Petermichl and Pipher [DPP] (in the `small constant’ case). These $L^p$ solvability results can be interpreted in terms of a reverse Hölder condition for the elliptic kernel and therefore connected with the $A_\infty$ condition. In this talk, we discuss L^p solvability for a class of coefficients that satisfies a `weak DKP condition’. In particular, we connect the (weak) DKP condition to the characterization of $A_\infty$ in [FKP]. This allows us to treat the `large’, `small’ and ‘vanishing’ (weak) DKP conditions simultaneously and independently from the works [KP] and [DPP].

This is joint work with my co-authors Egert, Saari, Toro and Zhao. A proof of the main estimate will be sketched, but technical details will be avoided.

https://sites.google.com/view/paw-seminar