Mnemosyne o la costruzione del senso n° 12 – 2019

In this thesis, we emphasise the role of a particular "integrable" structure in the study of determinantal point processes, namely kernels of integrable form.

We introduce an observation of a point process involving the notions of marketing and thinning, and then condition that point process on the resulting observation. This is shown to preserve important subclasses of point processes :

- determinantal point processes, i.e. point processes whose correlation functions can be written as determinants of a so-called correlation kernel: the transformation makes use of Fredholm determinants and minors ;

- determinantal point processes induced by kernels of projections ;

- determinantal point processes with kernels of integrable form : the transformation can be characterised by the solution to a Riemann-Hilbert problem (RHP).

We then study Jãnossy densities associated to observations of the Airy point process, which are, informally speaking, likelihoods of observations. We prove that we can apply the conditioning transformation to this process and that the kernel of the conditioned point process can be characterized by an RHP. Moreover :

• that RHP's solution satisfies linear differential equations and undergoes isomonodromic deformations ;
• from the compatibility solutions of these differential equations we obtain an isospectral deformation of the Stark operator whose potential is expressible in terms of the Jãnossy density and satisfies the cylindrical Kortreweg-de Vries equation ;
• we prove a trace formula implying that the Stark operator's eigenfunctions are solutions to an infinitely coupled version of the Painlevé II differential equation.

Gabriel Glesner obtained his Master in Mathematics in 2018 at UCLouvain. He then started a Ph.D in Mathematics under Pr. Tom Claeys's supervision within the GPP centre at UCLouvain, researching Riemann-Hilbert problems methods applied to integrable probabilistic models.