Given a certain invariant random matrix ensemble characterised by the joint probability distribution of eigenvalues P (λ 1 , . . . , λ N ), many important questions have been related to the study of linear statistics of eigenvaluesis a known function. We study here truncated linear statistics where the sum is restricted to the N 1 < N largest eigenvalues:Motivated by the analysis of the statistical physics of fluctuating one-dimensional interfaces, we consider the case of the Laguerre ensemble of random matrices with f (λ) = √ λ. Using the Coulomb gas technique, we study the N → ∞ limit with N 1 /N fixed. We show that the constraint that L = N1 i=1 f (λ 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. Assuming that f (λ) is monotonous, we show that these features arise for any random matrix ensemble and truncated linear statitics, which makes the scenario described here robust and universal. PACS numbers : 05.40.-a ; 02.50.-r ; 05.70.Np IntroductionIntroduced in physics by Wigner and Dyson in the 1950s in order to model the complexity in atomic nucleus, random matrix theory has irrigated many fields of physics, ranging from electronic quantum transport [1][2][3][4][5][6][7][8][9][10][11][12][13], quantum information (entanglement in random bipartite quantum states) [1. The case where the matrix distribution is invariant under changes of basis, i.e. when eigenvalues and eigenvectors are uncorrelated.2. In the paper, the expressions for the limiting behaviours of P N,κ (s) must be understood more rigorously as lim N →∞ − 2/(βN 2 ) ln[P N,κ (s)] = Φκ(s).