We prove a form of Arnold diffusion in the a-priori stable case. Let $$H_0(p)+\mathit{ϵ}{H}_1(\mathit{\theta },p,t),\mathit{\theta }\in {\mathbb{T}}^n,p\in B^n,t\in \mathbb{T}=\mathbb{R}/\mathbb{T},$$be a nearly integrable system of arbitrary degrees of freedom $n⩾2$ with a strictly convex H₀. We show that for a “generic” $\mathit{ϵ}{H}_1$, there exists an orbit $(\mathit{\theta },p)$ satisfying $$\Vert p(t)-p(0)\Vert >l(H_1)>0,$$where $l(H_1)$ is independent of $\mathit{ϵ}$. The diffusion orbit travels along a codimension-1 resonance, and the only obstruction to our construction is a finite set of additional resonances.

For the proof we use a combination of geometric and variational methods, and manage to adapt tools which have recently been developed in the a-priori unstable case.

We construct approximate transport maps for perturbative several-matrix models. As a consequence, we deduce that local statistics have the same asymptotic as in the case of independent GUE or GOE matrices (i.e., they are given by the sine-kernel in the bulk and the Tracy–Widom distribution at the edge), and we show averaged energy universality (i.e., universality for averages of m-points correlation functions around some energy level E in the bulk). As a corollary, these results yield universality for self-adjoint polynomials in several independent GUE or GOE matrices which are close to the identity.

We give lower bounds for the numbers of real solutions in problems appearing in Schubert calculus in the Grassmannian $\mathrm{Gr}(n,d)$ related to osculating flags. It is known that such solutions are related to Bethe vectors in the Gaudin model associated to ${\mathrm{gl}}_n$. The Gaudin Hamiltonians are self-adjoint with respect to a non-degenerate indefinite Hermitian form. Our bound comes from the computation of the signature of that form.