Bertrand Lacroix-A-Chez-Toine (King’s College London)
Superposition of random plane waves in high dimension as a random landscape
Superpositions of random plane waves play important roles in the semi-classical description of quantum billiard as pointed out by Berry’s conjecture . In this context, they can describe the eigenfunctions of the Laplacian operator at high energy. While their properties have been explored in depth in low spatial dimensions, we consider in this talk a large superposition of M ≫ 1 random plane waves in high dimension N ≫ 1 with M/N = α > 1. Here, we consider instead this object as a (random) energy landscape and, adding an isotropic harmonic confinement of strength μ, we characterise the ergodicity breaking in such a landscape. To characterise this property we consider two quantities: the quenched free energy and the annealed total complexity, i.e. the rate of exponential growth of the average number of stationary points of the energy landscape with the spatial dimension N. While similar high-dimensional random landscapes display topology trivialisation transition , whereby the complexity vanishes above some finite value of the confinement μ, the complexity vanishes only as μ → ∞ in this system . One might thus expect that ergodicity is always broken at zero temperature in this model. This is confirmed and enriched by our quenched free-energy computations.
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 B. Lacroix-A-Chez-Toine, S. Belga-Fedeli, Y. V. Fyodorov, Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity, J. Math. Phys. 63 (9), 093301 (2022)