Random polynomials, Coulomb gas, and random matrices.

Summary The main object of this thesis is the study of several models of random polynomials. The aim is to understand the macroscopic behavior of random polynomial roots whose degree tends to infinity. We will explore the connection between the roots of random polynomials and Coulomb gases in order to obtain large deviation principles for the empirical measurement of the roots. We revisit the paper of Zeitouni and Zelditch which establishes a large deviation principle for a general model of random polynomials with complex Gaussian coefficients. We extend this result to the case of real Gaussian coefficients. Then, we show that these results remain valid for a large class of laws on coefficients, making large deviations a universal phenomenon for these models. Moreover, we prove all the previous results for the model of renormalized Weyl polynomials. We are also interested in the behavior of the root of largest modulus of Kac polynomials. This one has a non-universal behavior and is in general a random variable with heavy tails. Finally, we prove a principle of large deviations for the empirical measurement of biorthogonal sets.