Timeline for Is there a version of Weyl's law for graph Laplacians?
Current License: CC BY-SA 4.0
11 events
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| Jul 22 at 20:45 | comment | added | Will Sawin | @Pulcinella. Each infinite graph has its own spectral measure. One can consider this spectral measure to be a "Weyl law", or one can consider the fact that the measure depends on the graph to be the absence of a 'Weyl law". For an infinite graph of bounded average edge degree, the spectrum is typically not infinitely large, so it's not clear what the right analogue of a Weyl law would be. | |
| Jul 22 at 8:26 | comment | added | Pulcinella | Should I take it from your answer that if one takes a particular (non-random) say infinite graph, there will not be a Weyl law for the eigenvalues? | |
| Jul 19 at 20:21 | history | bounty ended | Pulcinella | ||
| Jul 16 at 17:50 | comment | added | Gabe K | Your explanation concerning clusters does make sense for the graph of the Columbus block groups, since the particular node highlighted in the second picture has a large number of neighbors. | |
| Jul 16 at 17:42 | comment | added | Gabe K | Good point. Rereading Canzani's notes more carefully, she points out that "generically", the eigenfunctions are expected to behave like Gaussian random functions and have $L^\infty$-norm $O(\sqrt{\log \lambda})$. I think this would correspond to multiple peaks whose height is comparable, but the norm might still grow as the eigenvalue gets large. | |
| Jul 16 at 17:34 | comment | added | Will Sawin | @GabeK In neither case that I am familiar with is a typical high-frequency eigenfunction localized: In the sphere case these are very special eigenfunctions invariant under a large subgroup, and in the arithmetic hyperbolic case they are functorial transfers. So your colleague's observations still seem non-manifold-like to me. | |
| Jul 16 at 17:32 | comment | added | Will Sawin | @GabeK For a general curved manifold one expects the lower bound of the $L^\infty$ norm coming from the $L^2$ norm to be close to sharp. For locally symmetric spaces, one expects large $L^\infty$ norms to occur only for some global geometric reason. For the sphere, this is the global symmetry. For arithmetic hyperbolic manifolds there can be large values arising from special subgroups of the arithmetic group. In this case the points of the manifold where large values occur are special, it cannot be an arbitrary point. | |
| Jul 16 at 17:22 | comment | added | Gabe K | In general, the $L^\infty$ growth of the eigenfunctions is slower than that (the bound is only sharp in spherical geometry) and won't apply to every eigenfunction, but that's the intuition for why there can be sharp local peaks. | |
| Jul 16 at 17:19 | comment | added | Gabe K | Thanks for the helpful answer. About your last point, you are correct that for a uniform rectangular lattice, the high-frequency eigenvectors will not localize (and this might be true for grid graphs as well). However, in curved geometry, the $L^\infty$ norm of $L^2$-normalized eigenfunctions can grow like $O(\lambda^{(n-1)/4})$. My intuition is that a non-uniform grid should behave somewhat like a curved manifold, so this localization phenomena is reflecting that fact. | |
| Jul 16 at 14:24 | history | edited | Will Sawin | CC BY-SA 4.0 |
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| Jul 16 at 13:24 | history | answered | Will Sawin | CC BY-SA 4.0 |