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High-girth near-Ramanujan graphs with localized eigenvectors

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Abstract

We show that for every prime d and α ∈ (0, 1/6), there is an infinite sequence of (d + 1)-regular graphs G = (V, E) with high girth Ω(α logd(∣V∣), second adjacency matrix eigenvalue bounded by \((3/\sqrt 2)\sqrt d \), and many eigenvectors fully localized on small sets of size O(mα). This strengthens the results of [GS18], who constructed high girth (but not expanding) graphs with similar properties, and may be viewed as a discrete analogue of the “scarring” phenomenon observed in the study of quantum ergodicity on manifolds. Key ingredients in the proof are a technique of Kahale [Kah92] for bounding the growth rate of eigenfunctions of graphs, discovered in the context of vertex expansion and a method of Erdős and Sachs for constructing high girth regular graphs.

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Acknowledgement

We thank an anonymous referee for a very thorough reading and many helpful comments which improved the paper substantially.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544-1000, USA

    Noga Alon

  2. Department of Statistics, University of California at Berkeley, Berkeley, CA, 94720-3860, USA

    Shirshendu Ganguly

  3. Department of Mathematics, University of California at Berkeley, Berkeley, CA, 94720-3840, USA

    Nikhil Srivastava

Authors
  1. Noga Alon
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  2. Shirshendu Ganguly
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  3. Nikhil Srivastava
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Corresponding author

Correspondence to Nikhil Srivastava.

Additional information

This work was partially completed while SG and NS were at the Simons Institute for the Theory of Computing, as part of the “Geometry of Polynomials” program.

Research supported in part by NSF grant DMS-1855464, ISF grant 281/17, BSF grant 2018267 and the Simons Foundation.

Partially supported by a Sloan Research Fellowship in mathematics and NSF Award DMS-1855688.

Supported by NSF grant CCF-1553751.

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Alon, N., Ganguly, S. & Srivastava, N. High-girth near-Ramanujan graphs with localized eigenvectors. Isr. J. Math. 246, 1–20 (2021). https://doi.org/10.1007/s11856-021-2217-y

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  • DOI: https://doi.org/10.1007/s11856-021-2217-y

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