Truncated linear statistics associated with the eigenvalues of random matrices

Jeudi 16 novembre 2017 15:30-16:30 - Christophe Texier - Université Paris-Sud

Lieu : salle 117/119 du bâtiment 425

Notes de dernières minutes : Given a certain invariant random matrix ensemble characterised by the joint probability distribution of eigenvalues $P(\lambda_1,\cdots,\lambda_N)$, the study of linear statistics of the eigenvalues $L=\sum_{i=1}^N f(\lambda_i)$, where $f(\lambda)$ is a known function, has played an important role in many applications of random matrix theory. I will discuss the distribution of truncated linear statistics of the form $\tilde{L}=\sum_{i=1}^{N_1} f(\lambda_i)$, when the sum runs over a fraction of the eigenvalues ($N_1<N$). By using the Coulomb gas technique, the large deviation function controlling the distribution of such sums in the limit of large $N$, with $0 < N_1/N < 1$ fixed, will be analysed. Two situations will be considered leading to two different universal scenarii : -# the case where the truncated linear statistics is restricted to the largest (or smallest) eigenvalues. We have shown that the constraint that $\tilde{L}=\sum_{i=1}^{N_1} f(\lambda_i)$ is fixed drives an infinite order phase transition in the underlying Coulomb gas. This transition corresponds to a change in the density of the gas, from a density defined on two disjoint intervals to a single interval. In this latter case the density presents a logarithmic divergence inside the bulk. -# the second situation is the case without further restriction on the ordering of the eigenvalues contributing to the truncated linear statistics (this can be viewed as a new ensemble which is related, but not equivalent, to the ``thinned ensembles’’ introduced by Bohigas and Pato). In this case, a region opens in the phase diagram of the Coulomb gas, where the large deviation function is mostly controlled by entropy (in particular this induces a change in the scaling of the relative fluctuations of the truncated linear statistics, from the usual $1/N$ for $N_1=N$, to $1/\sqrt{N}$ when $N_1<N$). Our analysis relies on the mapping on a problem of $N_1$ fictitious non-interacting fermions in $N$ energy levels, which can exhibit both positive and negative effective (absolute) temperatures.