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Many Nodal Domains in Random Regular Graphs

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Abstract

Let G be a random d-regular graph on n vertices. We prove that for every constant \(\alpha > 0\), with high probability every eigenvector of the adjacency matrix of G with eigenvalue less than \(-2\sqrt{d-2}-\alpha \) has \(\Omega (n/\textrm{polylog}(n))\) nodal domains.

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Notes

  1. We restrict our attention to weak nodal domains as there are at least as many strong domains as weak domains.

  2. The quantum chaos literature considers eigenfunctions of Laplacians on manifolds in the high energy limit. We analogously consider the highest Laplacian eigenvalues of graphs of fixed degree with the number of vertices going to infinity. High eigenvalue eigenfunctions have high frequency, and therefore, by the Planck formula, high energy.

  3. As a starting point, Eldan, H. Huang, and Rudelson asked in 2020 [Rud20] whether the most negative eigenvector of a sparse G(n, p) graph has more than two nodal domains. Such graphs may have nontrivial nodal domain structure when p is near or below the connectivity threshold \(\log n/n\). However, these graphs have a different, irregular, Benjamini-Schramm limit from random \(d-\)regular graphs. We expect our methods to work in this new setting, but this would require significant modifications, and we have not pursued this in this work.

  4. The Green’s function \((A-zI)^{-1}\) of a random regular graph can only approximate that of the infinite tree when \(\Im (z)\geqslant \textrm{polylog}n/n\), meaning that it inherently reflects the aggregate behavior of \(\textrm{polylog}n\) eigenvectors.

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Authors and Affiliations

  1. UC Berkeley, Berkeley, USA

    Shirshendu Ganguly, Sidhanth Mohanty & Nikhil Srivastava

  2. Harvard University, Cambridge, USA

    Theo McKenzie

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Correspondence to Shirshendu Ganguly.

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Communicated by J. Ding

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Shirshendu Ganguly supported by NSF Grant DMS-1855688, NSF Career Grant DMS-1945172, and a Sloan Fellowship.

Theo McKenzie supported by NSF GRFP Grant DGE-1752814 and NSF Grant DMS-2212881.

Sidhanth Mohanty supported by Google Ph.D. Fellowship.

Nikhil Srivastava supported by NSF Grant CCF-2009011.

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Ganguly, S., McKenzie, T., Mohanty, S. et al. Many Nodal Domains in Random Regular Graphs. Commun. Math. Phys. 401, 1291–1309 (2023). https://doi.org/10.1007/s00220-023-04709-6

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