Critical sets of random smooth functions on compact manifolds

Given a compact, connected Riemann manifold without boundary $(M, g)$ of dimension $m$ and a large positive constant $L$ we denote by $U_L$ the subspace of $C^{\infty}(M)$ spanned by eigenfunctions of the Laplacian corresponding to eigenvalues $\leq L$. We equip $U_L$ with the standard Gaussian probability measure induced by the $L^2$-metric on $U_L$, and we denote by $\mathcal{N}_L$ the expected number of critical points of a random function in $U_L$. We prove that $\mathcal{N}_L \sim C_m \dim U_L$ as $L \to \infty$, where $C_m$ is an explicit positive constant that depends only on the dimension $m$ of $M$ and satisfying the asymptotic estimate $\log C_m \sim \frac{m}{2} \log m$ as $m \to \infty$.

Keywords

random Morse functions, critical points, Kac-Rice formula, gaussian random processes, spectral function, random matrices

2010 Mathematics Subject Classification

Primary 15B52. Secondary 42C10, 53C65, 58K05, 60D05, 60G15, 60G60.

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Published 19 June 2015