Guillaume Dubach - Courant Institute, NYU
Courant Institute
Guillaume DUBACH


Powers of Ginibre Eigenvalues
(Prepublished on November 8th, 2017.)

The distribution of overlaps between eigenvectors of Ginibre matrices
(Joint work with P. Bourgade. Prepublished on January 3rd, 2018.)
This video illustrates one aspect of our results.


I gave a Junior Participant talk at the 2017 North-East Probability Seminar.
(November 16th).

Title: The overlaps between Ginibre eigenvectors

Abstract : The Complex Ginibre Ensemble consists of matrices whose coefficients are i.i.d. complex Gaussian. The eigenvectors are non-orthogonal, and their distance to a unitary basis can be quantified through the matrix of overlaps. These variables quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of the overlaps and their correlations. (Joint work with P. Bourgade)


FALL 2017

The Student Probability Seminar happens on wednesdays, 10am to 11am, usually in room 1314.

Student Probability Seminar, october 18th.
At 10am in room WWH 1314.
Title : Self-Avoiding Walks on the Honeycomb Lattice.
Abstract :
While in general there is no exact formula that counts self-avoiding walks on a given infinite graph, the number of self-avoiding walks of length k on lattices is known to be logarithmically equivalent to ck , where c is called the connective constant of the lattice, and usually can only be approximated. In the specific case of the honeycomb lattice (i.e. the hexagonal structure one finds in beehives), physical heuristics led to the conjecture that the connective constant is
Connective Constant
Duminil-Copin and Smirnov proved this conjecture a few years ago in a sensational thirteen-page-long paper. We will sketch their proof, discuss why it cannot be easily extended to other settings, and what else can be asked or expected from a probabilistic point of view.

Self Avoiding Walks


Student Probability Seminar, april 3rd.
At 11am in room WWH 905.
Title : Ginibre Powers - the Laundry Machine Effect.
Abstract :
The name of Jean Ginibre was given to a very natural ensemble of random matrices, those with iid complex gaussian coefficents. The eigenvalues of such a matrix form a determinantal point process, and thus exhibit a well-studied repulsion. But this highly correlated system of points happens to decorrelate completely (I say : completely - not asymptotically) when put to a high enough power - that is, having them spin around the origin. This stunning property is known to hold for a wider class of processes; but the proof is particularly straightforward in the Ginibre case, as we shall see.

Student Probability Seminar, march 6th.
At 11am in room WWH 102.
Title : Some problems of random cycles - and how to fix them.
Abstract :
We shall consider the uniform distribution over the permutation groups and try to answer simple questions such as : how likely is it for two elements to belong to the same cycle ? How are fixed points distributed ? What is the typical size of a cycle ?... Some of our answers will be just as simple as the questions; whereas others will require to introduce usual tools and methods of random permutation theory, such as the celebrated Feller coupling, thanks to which we might even end up answering questions we didn't ask.

Random Cycles Talk

FALL 2016

Student Probability Seminar, october 25th.
At 2pm in room WWH 1314.
Title : The Characteristic Polynomial of a Random Unitary Matrix.
Abstract :
We will introduce the Circular Unitary Ensemble (CUE) of random unitary matrices distributed according to the Haar measure on Un(C), and compute the distribution of its characteristic polynomial thanks to an explicit decomposition of the law. In particular, the moments of this characteristic polynomial can be computed ; they are strongly believed to have a link with the moments of the zeta function along its critical line, and we will give some reasons why this might be true.


Student Probability Seminar, february 25th.
At 4pm in room WWH 1314.
Title : Mere coincidences, or perhaps more.
Abstract :
The story of the meeting at tea time between Freeman Dyson and Hugh Montgomery, and what happened that the latter would later recall as 'real serendipity', is now famous; but a good story never bores. We shall tell it again, sketch its mathematical background, and present a few other number-theoretic objects that exhibit amazing similarities with well-known results in Random Matrix theory : not only the zeros of the Zeta function, but also its moments on the critical line, andthe number of points of some elliptic curves over finite fields.

FALL 2015

Student Probability Seminar, october 29th, 2015.
Title : Around Ulam's problem.
Abstract :
This talk will recall and sketch famous results about the longest
increasing subsequence of a random (uniformly chosen) permutation. For this
purpose,we shall learn and play a solitaire game, give a probabilistic proof of two elegant hook-formulas, and explain the Robinson-Schensted algorithm, among other things. These tools will give us two different approaches of the same problem, and even allow us to draw some conclusions.

I took and passed my Oral examination on october 13th.
Here is my outline.


My Master's Thesis "Approches probabilistes de la fonction Zêta" will be displayed online one day.

My Mémoire de Maîtrise, "Corps de u-invariant pair" was co-written with Margaret Bilu. The u-invariant of a field k is the maximal dimension of an anisotropic quadratic form on k. It was conjectured at some point that it should be a power of 2 for every field; which is not the case. In this paper, following an argument given by A. Merkurjev, we build a field of u-invariant 2n for any positive integer n.

Furious Researcher
Updated: 01/03/2018