Short geodesic loops and $$L^p$$ L p norms of eigenfunctions on large genus random surfaces

@article{Gilmore2019ShortGL,
  title={Short geodesic loops and 
 
 
 
 \$\$L^p\$\$
 
 
 L
 p
 
 
 norms of eigenfunctions on large genus random surfaces},
  author={Clifford Gilmore and Etienne Le Masson and Tuomas Sahlsten and Joe Thomas},
  journal={Geometric and Functional Analysis},
  year={2019},
  volume={31},
  pages={62-110}
}
We give upper bounds for $$L^p$$ L p norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus $$g \rightarrow +\infty $$ g → + ∞ , we show that random hyperbolic surfaces X with respect to the Weil-Petersson volume have with high probability at most one such loop of length less than $$c \log g$$ c log g for small enough $$c > 0$$ c > 0 . This allows… Expand

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References

SHOWING 1-10 OF 55 REFERENCES
Improvement of eigenfunction estimates on manifolds of nonpositive curvature
Let $(M,g)$ be a compact, boundaryless manifold of dimension $n$ with the property that either (i) $n=2$ and $(M,g)$ has no conjugate points, or (ii) the sectional curvatures of $(M,g)$ areExpand
Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces
  • A. Saha
  • Mathematics
  • Mathematische Annalen
  • 2019
Let D be an indefinite quaternion division algebra over $${{\mathbb {Q}}}$$ Q . We approach the problem of bounding the sup-norms of automorphic forms $$\phi $$ ϕ on $$D^\times ({{\mathbb {A}}})$$ DExpand
On the almost eigenvectors of random regular graphs
Let $d\geq 3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices. We show that if $n$ is large enough then the entry distribution of every almost eigenvector $v$ of $G$ (with entryExpand
On the sup-norm of Maass cusp forms of large level
We establish upper bounds for the sup-norm of Hecke-Maass eigenforms on arithmetic surfaces. In a first part, the case of open modular surfaces is studied. Let $${f}$$ be an Hecke–Maass cuspidalExpand
$L^p$ Norms of Eigenfunctions on Regular Graphs and on the Sphere
We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian andExpand
On the sup-norm of Maass cusp forms of large level. III
Let $$f$$ be a Hecke–Maass cuspidal newform of square-free level $$N$$ and Laplacian eigenvalue $$\lambda $$. It is shown that $$\left||f \right||_\infty \ll _{\lambda ,\epsilon }Expand
On the growth of $L^2$-invariants for sequences of lattices in Lie groups
We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of theExpand
$L^\infty$ norms of eigenfunctions of arithmetic surfaces
Here and elsewhere A ,l1o0 as A -xc is a basic problem of Quantum Chaos [S]. In any case almost nothing beyond (0.2) is known about 110,joc, when the curvature is negative (one can push the standardExpand
Local Kesten–McKay Law for Random Regular Graphs
We study the adjacency matrices of random d-regular graphs with large but fixed degree d. In the bulk of the spectrum $${[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]}$$[-2d-1+ε,2d-1-ε] down toExpand
Ergodic billiards that are not quantum unique ergodic
Partially rectangular domains are compact two-dimensional Riemannian manifolds $X$, either closed or with boundary, that contain a flat rectangle or cylinder. In this paper we are interested inExpand
...
1
2
3
4
5
...