740 research outputs found

    Random matrices in non-confining potentials

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    We consider invariant matrix processes diffusing in non-confining cubic potentials of the form Va(x)=x3/3ax,aRV_a(x)= x^3/3 - a x, a\in \mathbb{R}. We construct the trajectories of such processes for all time by restarting them whenever an explosion occurs, from a new (well chosen) initial condition, insuring continuity of the eigenvectors and of the non exploding eigenvalues. We characterize the dynamics of the spectrum in the limit of large dimension and analyze the stationary state of this evolution explicitly. We exhibit a sharp phase transition for the limiting spectral density ρa\rho_a at a critical value a=aa=a^*. If aaa\geq a^*, then the potential VaV_a presents a well near x=ax=\sqrt{a} deep enough to confine all the particles inside, and the spectral density ρa\rho_a is supported on a compact interval. If a<aa<a^* however, the steady state is in fact dynamical with a macroscopic stationary flux of particles flowing across the system. In this regime, the eigenvalues allocate according to a stationary density profile ρa\rho_{a} with full support in R\mathbb{R}, flanked with heavy tails such that ρa(x)Ca/x2\rho_{a}(x)\sim C_a /x^2 as x±x\to \pm \infty. Our method applies to other non-confining potentials and we further investigate a family of quartic potentials, which were already studied in Br\'ezin et al. to count planar diagrams.Comment: 32 pages, 7 figure

    Tracy-Widom at high temperature

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    We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature β\beta tends to 00. We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy-Widom β\beta law, converges weakly, when properly centered and scaled, to the Gumbel distribution. More generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index kk. Those convergences are obtained after a careful analysis of the explosion times process of the Riccati diffusion associated to the stochastic Airy operator. We show that the empirical measure of the explosion times converges weakly to a Poisson point process using estimates proved in [L. Dumaz and B. Vir\'ag. Ann. Inst. H. Poincar\'e Probab. Statist. 49, 4, 915-933, (2013)]. We further compute the empirical eigenvalue density of the stochastic Airy ensemble on the macroscopic scale when β0\beta\to 0. As an application, we investigate the maximal eigenvalues statistics of βN\beta_N-ensembles when the repulsion parameter βN0\beta_N\to 0 when N+N\to +\infty. We study the double scaling limit N+,βN0N\to +\infty, \beta_N \to 0 and argue with heuristic and numerical arguments that the statistics of the marginal distributions can be deduced following the ideas of [A. Edelman and B. D. Sutton. J. Stat. Phys. 127, 6, 1121-1165 (2007)] and [J. A. Ram\'irez, B. Rider and B. Vir\'ag. J. Amer. Math. Soc. 24, 919-944 (2011)] from our later study of the stochastic Airy operator.Comment: 5 figure

    A diffusive matrix model for invariant β\beta-ensembles

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    We define a new diffusive matrix model converging towards the β\beta-Dyson Brownian motion for all β[0,2]\beta\in [0,2] that provides an explicit construction of β\beta-ensembles of random matrices that is invariant under the orthogonal/unitary group. We also describe the eigenvector dynamics of the limiting matrix process; we show that when β<1\beta< 1 and that two eigenvalues collide, the eigenvectors of these two colliding eigenvalues fluctuate very fast and take the uniform measure on the orthocomplement of the eigenvectors of the remaining eigenvalues

    Invariant β\beta-ensembles and the Gauss-Wigner crossover

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    We define a new diffusive matrix model converging towards the β\beta -Dyson Brownian motion for all β[0,2]\beta\in [0,2] that provides an explicit construction of β\beta-ensembles of random matrices that is invariant under the orthogonal/unitary group. For small values of β\beta, our process allows one to interpolate smoothly between the Gaussian distribution and the Wigner semi-circle. The interpolating limit distributions form a one parameter family that can be explicitly computed. This also allows us to compute the finite-size corrections to the semi-circle.Comment: 3 figure
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