Survey on spectral gaps of random hyperbolic surfaces.
problem Understanding spectral gaps of random hyperbolic surfaces.
method Brief survey on geometry and spectra, discussion of results by Hide-Magee, Anantharaman-Monk, and Hide-Macera-Thomas.
result Near optimal spectral gaps for random surfaces.
Random hyperbolic surfaces have low Cheeger constants.
problem Estimating Cheeger constants of random hyperbolic surfaces.
method Modeling random hyperbolic surfaces using ideal triangles and analyzing their Cheeger constants.
result Generic hyperbolic surfaces have Cheeger constants less than 3/2π + ε.
Random hyperbolic surfaces have nearly optimal spectral gaps.
problem Proving the nearly optimal spectral gap conjecture for random Belyi surfaces.
method Using the Brooks-Makover model, the authors show a spectral gap greater than 1/4 - c/log(n).
result A random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than 1/4 - c/log(n).
The study finds arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps.
problem Understanding spectral gaps of random hyperbolic surfaces with many cusps.
method Analysis of moduli spaces of hyperbolic surfaces with Weil-Petersson metric.
result Arbitrarily small spectral gaps are observed as the number of cusps grows slower than the genus.
Study reveals uniform spectral gaps for random hyperbolic surfaces with few cusps.
problem Investigating spectral gaps for random hyperbolic surfaces with limited cusps.
method Analyzing Weil-Petersson random hyperbolic surfaces, showing no eigenvalues in specific intervals.
result Uniform lower bounds on spectral gaps for Weil-Petersson random hyperbolic surfaces, revealing a critical phenomenon of 'second order cancellation'.
Study shows exponential growth of Laplacian determinant on random hyperbolic surfaces.
problem Understanding the behavior of Laplacian determinants on random hyperbolic surfaces.
method Investigated various models of random hyperbolic surfaces and their Laplacian determinants as genus increases.
result For all popular models, the determinant grows exponentially with a universal exponent as the genus goes to infinity.
Random hyperbolic surfaces are mostly tangle-free, with geometric implications.
problem Understanding the structure of random hyperbolic surfaces.
method Introduced and analyzed L-tangle-free compact hyperbolic surfaces.
result Random surfaces are (a log g)-tangle-free for any a < 1, almost optimal.
We determine the asymptotic growth rate of the diameter of the random hyperbolic surfaces constructed by Brooks and Makover. This model consists of a uniform gluing of 2n hyperbolic ideal triangles along their sides followed by a compactification to get a random hyperbolic surface of genus roughly n/2. We show that…
Random covers of hyperbolic surfaces have a spectral gap with polynomial rate.
problem Finding spectral gaps in random covers of hyperbolic surfaces.
method Applying recent work on spectral gaps to uniformly random covers of closed hyperbolic surfaces.
result Uniformly random degree-n covers of a closed hyperbolic surface have no new Laplacian eigenvalues below a specific threshold with high probability.
Study on random surfaces in hyperbolic 3-manifolds, focusing on geometric and topological properties.
problem Distribution of nearly geodesic surfaces in hyperbolic 3-manifolds.
method Invariant measures on the Grassmann bundle G(M) derived from limits of random minimal surfaces.
result Topological limiting measures are totally scarring if M contains a totally geodesic subsurface, while geometrical limiting measures are not.
Random hyperbolic surfaces with punctures converge to the Brownian sphere.
problem Understanding the geometry of random hyperbolic surfaces with punctures.
method Rescaling and encoding via plane trees with continuous labels.
result Rescaled random hyperbolic surfaces converge to the Brownian sphere.
Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.
problem Distribution of shapes of complementary subsurfaces in moduli space.
method Study of shapes of complementary subsurfaces in moduli space as boundary lengths go to infinity.
result Random subsurfaces look like random ribbon graphs.
Random surfaces with long systoles created from graph theory ideas.
problem Finding surfaces with long systoles.
method Two constructions inspired by graph theory.
result Proved a new lower bound on systole length.
The length of shortest non-simple geodesics grows logarithmically with surface genus.
problem Understanding the behavior of shortest non-simple closed geodesics on hyperbolic surfaces.
method Investigation of asymptotic behavior on random hyperbolic surfaces using the Weil-Petersson measure.
result The non-simple systole behaves like log(g) as g goes to infinity.
Study on random hyperbolic surfaces with many cusps, focusing on tight geodesics.
problem Understanding length statistics of geodesics on random hyperbolic surfaces with cusps.
method Recursion formula for tight Weil-Petersson volumes and generalization of Mirzakhani's integration formula.
result Recovery of Poisson point process in large genus limit for length statistics of tight geodesics.
Eigenvalues of random hyperbolic surface covers converge to hyperbolic plane's.
problem Eigenvalue rigidity of random hyperbolic surface covers.
method Selberg trace formula and polynomial method.
result Distribution of eigenvalues converges to hyperbolic plane's spectral measure.
We prove Poisson approximation results for the bottom part of the length spectrum of a random closed hyperbolic surface of large genus. Here, a random hyperbolic surface is a surface picked at random using the Weil-Petersson volume form on the corresponding moduli space. As an application of our result, we compute the …
Study on geodesics and eigenvalues on random hyperbolic surfaces with cusps.
problem Counting short geodesics and small eigenvalues on random hyperbolic surfaces.
method Rescaling and convergence to a Poisson point process.
result The probability of having at least k=o(n) arbitrarily small eigenvalues tends to 1 as no∞. Random covers of hyperbolic surfaces follow a specific probability measure.
problem Understanding the distribution of random covers of hyperbolic surfaces.
method Analyzing random covers subject to specific group isomorphism conditions.
result Asymptotic distribution of random covers according to a probability measure on moduli space of metric graphs.
Study shows energy levels on hyperbolic surfaces follow GOE fluctuations.
problem Understanding energy level fluctuations on hyperbolic surfaces.
method Analysis of Laplace eigenvalues on hyperbolic surfaces, using GOE random matrix theory.
result Energy variance on typical hyperbolic surfaces closely matches GOE fluctuations.
Random hyperbolic surfaces have a spectral gap that approaches 1/4 as genus grows.
problem Estimating the spectral gap of random hyperbolic surfaces.
method Analyzing the Weil-Petersson measure on moduli spaces of metrics.
result The spectral gap of random hyperbolic surfaces converges to 1/4 as the genus increases.
New upper bound for Cheeger constant of hyperbolic surfaces.
problem Bounding the Cheeger constant of hyperbolic surfaces.
method Random construction based on Poisson--Voronoi tessellation.
result The Cheeger constant of closed hyperbolic surfaces is less than that of the hyperbolic plane.
Study on geodesics on random hyperbolic surfaces, showing variance asymptotic to X log X.
problem Distribution of closed geodesics on random hyperbolic surfaces.
method Viewing surfaces as random points in moduli space, studying weighted counting function.
result Variance in large genus limit is asymptotic to X log X, with exceptions.
Random surfaces have a strong spectral gap with polynomial rate.
problem Understanding spectral gaps in random hyperbolic surfaces.
method Adapting polynomial method for random matrices to Laplacian on surfaces.
result Laplacian spectral gap at least 1/4 - O(1/g^c) for large g.
In this note, we prove that a random extension of either the free group FN of rank N≥3 or of the fundamental group of a closed, orientable surface Sg of genus g≥2 is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either Out(FN) or Mod(Sg) generated by k independ…
Study shows systole behavior changes significantly for large genus hyperbolic surfaces.
problem Understanding systole behavior in large genus hyperbolic surfaces.
method Analysis of random surfaces with respect to Weil-Petersson volume.
result Expected value of separating systole behaves like 2logg for large genus. The study proves optimal spectral gaps for hyperbolic surfaces.
problem Proving optimal spectral gaps for hyperbolic surfaces.
method Proving the absence of eigenvalues in a specific range for random covers of hyperbolic surfaces.
result The first non-zero eigenvalue of the Laplacian on a sequence of closed hyperbolic surfaces tends to 1/4.
New model calculates logarithmic surface diameter.
problem Calculating diameter of random hyperbolic surfaces.
method Exploration process inspired by graph breadth-first search.
result Diameter is logarithmic in surface genus.
Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.
problem Analyzing critical exponents on hyperbolic surfaces with long boundaries.
method Using spine graph construction and comparing normalized Weil-Petersson and Kontsevich measures.
result Asymptotic convergence-in-mean result of normalized Weil-Petersson measures to normalized Kontsevich measures.
The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.
problem Finding the asymptotic behavior of shortest closed multi-geodesics on hyperbolic surfaces.
method Analyzing the length of shortest filling closed multi-geodesics using hyperbolic geometry and asymptotic analysis.
result The length of a shortest filling closed multi-geodesic is uniformly comparable to a specific formula involving the genus and lengths of closed geodesics.
We prove that the minimal diameter of a hyperbolic compact orientable surface of genus g is asymptotic to logg as g→∞. The proof relies on a random construction, which we analyse using lattice point counting theory and the exploration of random trivalent graphs.
The paper constructs noncompact hyperbolic surfaces with uniform spectral gaps using random graph models.
problem Building noncompact hyperbolic surfaces with uniform spectral gaps.
method Introduced a random graph model Fχ,n to construct expanding families of graphs, then applied these families to create hyperbolic surfaces. result Explicitly constructed an expanding family of graphs in the critical regime, leading to a sequence of complete, noncompact hyperbolic surfaces with uniformly positive spectral gaps.
Maps with a single face converge to hyperbolic surfaces in large genus.
problem Understanding geometric properties of high genus maps.
method Analyzing uniformly random maps and their convergence to hyperbolic surfaces.
result Lengths of simple cycles converge to a Poisson process.
Formula removes geometric patterns from random hyperbolic surfaces.
problem Conditioning on tangle-free surfaces to avoid rare geometric patterns.
method Developed a Moebius inversion formula to integrate tangle-free surfaces.
result Significantly reduces the number of local topological types of short geodesics.
The study finds effective lower bounds for spectra of random surfaces and bundles.
problem Determining the spectrum of Laplacian on random surfaces and bundles.
method Analysis of random covering surfaces and unitary bundles over finite-area non-compact hyperbolic surfaces.
result With high probability, the spectrum of random surfaces and bundles has no eigenvalues below a certain threshold.
The paper shows Gaussian fluctuations in eigenvalue statistics of random hyperbolic surfaces.
problem Understanding fluctuations in Laplace eigenvalues of random hyperbolic surfaces.
method Analyzing fluctuations of linear statistics of Laplace eigenvalues over moduli space of surfaces of large genus.
result The distribution of linear statistics tends to a Gaussian as the genus of surfaces increases.
Study smooth linear statistics on random covers of hyperbolic surfaces, showing central limit and variance results.
problem Analyzing fluctuations and energy variance of random covers of compact hyperbolic surfaces.
method Examining fluctuations in a small energy window around a fixed energy level, considering the variance of a typical surface, using a double limit where n and L go to infinity. result Distribution of fluctuations tends to a Gaussian with variance of GOE/GUE, and energy variance of a typical random n-cover is that of GOE/GUE. Study of large-n asymptotics for Weil-Petersson volumes of hyperbolic surfaces with cusps.
problem Understanding the geometry and spectral properties of random hyperbolic surfaces with many cusps.
method Large-n asymptotic analysis, spectral theory, and moduli space volumes. result Linear number of small Laplacian eigenvalues and relative frequency of simple vs. non-simple closed geodesics.
Directly proves logarithmic systolic growth for all hyperbolic surfaces.
problem Proving logarithmic systolic growth for all hyperbolic surfaces.
method Using original Brooks/Buser-Sarnak surfaces through a direct approach.
result Directly proves logarithmic systolic growth for all hyperbolic surfaces.
Study geodesics on random hyperbolic surfaces, finding variance similar to prime number theory.
problem Distribution of closed geodesics on random hyperbolic surfaces.
method Investigate random variable counting geodesics with norms in short intervals, comparing to prime number theory.
result Establishes variance of geodesic counting function is asymptotic to \(2H \log X\).
The study proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.
problem Proving nearly geodesic surfaces are filling in hyperbolic 3-manifolds.
method Using quasi-Fuchsian surfaces and totally geodesic surfaces, the study proves filling properties with rigorous mathematical proofs.
result Proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.
Our goal is to show, in two different contexts, that "random" surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition requires curves of total length at least g7/6−ε. Moreover, we prove that this bound holds for most metrics in the…
Formula for integrating random variables on hyperbolic surfaces.
problem Integrating random variables on the moduli space of hyperbolic surfaces.
method Integration formula for lengths of closed geodesics.
result Integral of geometric random variables can be expressed as an integral over R.
In this paper, we determine the distribution of the length partition of a random multicurve of fixed topological type on a closed hyperbolic surface using the methods of Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres. This distribution admits a polynomial density, whose coefficients can be e…
Study on length distribution of random multicurves on large genus surfaces converging to Poisson-Dirichlet distribution.
problem Length statistics of random multicurves on large genus hyperbolic surfaces.
method Analytical proof of convergence to Poisson-Dirichlet distribution as genus tends to infinity.
result Mean lengths of the three longest components converge to specific percentages of total length as genus increases.
New volume functions for random hyperbolic surfaces link to spectral gaps.
problem Analyzing spectral gaps in random hyperbolic surfaces.
method Introduced new volume functions VgT(l), derived their asymptotic expansions, and linked them to spectral gaps. result Coefficients in the asymptotic expansion of VgT(l) are Friedman-Ramanujan functions. The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric. Given suitable restrictions on the genus of the surface, we consider the number…
For large genus hyperbolic surfaces, this paper proves eigenvalue conditions and diameter bounds.
problem Eigenvalue and diameter bounds for large genus hyperbolic surfaces.
method Analysis of moduli space of hyperbolic surfaces with Weil-Petersson metric.
result Generic hyperbolic surfaces of large genus have first eigenvalues greater than 3/16 - ε.