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Description:
We consider invariant matrix processes diffusing in non-confining cubic potentials of the form $V_a(x)= x^3/3 - a x, a\in \mathbb {R}$. 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 $\rho _a$ at a critical value $a=a^*$. If $a\ge a^*$, then the potential $V_a$ presents a well near $x=\sqrt{a}$ deep enough to confine all the particles inside, and the spectral density $\rho _a$ is supported on a compact interval. If $a<a^*$ however, the steady state is in fact dynamical with a macroscopic stationary flux of particles flowing across the system. We prove that this flux $j_a$ displays a second order phase transition at the critical value $a^*$ such that $j_a\sim C (a^*-a)^{3/2}$ when a\uparrow $a^*$ where $C$ is an explicit constant. In the subcritical regime, the eigenvalues allocate according to a stationary density profile $\rho _{a}$ with full support in $\mathbb {R}$, flanked with heavy tails such that $\rho _{a}(x)\sim C_a /x^2$ as $x\rightarrow \pm \infty$ . Our method applies to other non-confining potentials and we further investigate a family of quartic potentials, which were already studied in (Brezin et al. in Commun Math Phys 59:35–51, 1978) to count planar diagrams.