Sharp spectral gap estimates on manifolds with integral curvature bounds.
problem Proving spectral gap estimates on manifolds with integral curvature bounds.
method Generalizing previous results to include integral curvature bounds.
result Confirms a conjecture about spectral gap estimates on manifolds with integral curvature bounds.
New estimates show spectral gap stability in RCD spaces, close to Beta distribution.
problem Stability of spectral gap bounds in metric-measure spaces.
method Combines L1-functional inequality and Stein's method. result Sharp quantitative estimate for spectral gap stability.
Estimates spectral gap for sub-Laplacian on compact manifolds.
problem Bounding the spectral gap of sub-Laplacian on compact manifolds.
method Lichnerowicz estimate adaptation for sub-Laplacian on compact manifolds.
result A bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian.
A new algorithm estimates spectral gap of Markov chains efficiently.
problem Estimating the spectral gap of a Markov chain efficiently.
method UCPI (Upper Confidence Power Iteration) algorithm
result Estimates spectral gap in O(n) time and O((lnn)2) memory. Estimates spectral gap for Brownian motion on sticky-reflecting domains.
problem Estimating spectral gap for Brownian motion on sticky-reflecting domains.
method Interpolation method and novel applications of Reilly formula.
result Lower bounds for spectral gap derived for general domains.
Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.
problem Estimating spectral gaps for higher-order operators on Cartan-Hadamard manifolds.
method Symmetrization-free proofs based on general functional inequalities.
result Solves a sharp asymptotic problem from Cheng and Yang and answers a question from Kristály.
Proves spectral gap bounds for Teichmüller geodesics on flat surfaces.
problem Quantify spectral gaps for Teichmüller geodesics.
method Bounding spectral gaps in terms of geometric quantities on flat surfaces.
result Quantitative non-uniform hyperbolicity of Teichmüller geodesic flow.
In this paper, we give an easy proof of the main results of Andrews and Clutterbuck's paper [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916], which gives both a sharp lower bound for the spectral gap of a Schröinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We…
Uniform spectral gap found for stable commutator length in hyperbolic 2-orbifolds.
problem Understanding stable commutator length in 3-manifolds.
method Explicit quasimorphisms for generic case, hyperbolic geometry for exceptional case.
result Explicit uniform gap of 1/36 for all orbifolds except a sphere with three cone points.
Paper generalizes Bakry-Émery calculus for curvature and applies to Markov chains.
problem Formulating both Bakry-Émery and entropic curvature simultaneously.
method Generalization of Bakry-Émery calculus, new measure optimality criterion, dimension parameter in entropic curvature.
result Diameter estimates for Markov chains with strictly positive entropic curvature and spectral gap.
Estimates Markov chain mixing time from a single trajectory.
problem Estimating mixing time of Markov chains from a single trajectory.
method Contraction with respect to total variation, inspired by Wolfer's contraction coefficient.
result Improved confidence intervals and instance-dependent rates for estimating Markov chains.
In the previous work [35], the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen-Wang [9, 10] and Bakry-Qian [6], from smoot…
The article proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
problem Proving a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
method Establishing conformal log-concavity estimates for the first eigenfunction.
result Proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
Estimates spectral gap from single Markov chain sample path.
problem Estimating spectral gap of unknown reversible Markov chains from a single trajectory.
method Uses relaxation time and stationary distribution properties to construct a data-dependent interval for mixing time.
result First procedure for estimating mixing time with high probability and without prior knowledge.
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).
We study the spectral gap of the Erdős--Rényi random graph through the connectivity threshold. In particular, we show that for any fixed δ>0 if p≥n(1/2+δ)logn, then the normalized graph Laplacian of an Erdős--Rényi graph has all of its nonzero eigenvalues tightly concentrated around 1. We est…
Estimates covariance matrices using Markov chain Monte Carlo with improved sample complexity.
problem Complexity of covariance matrix estimation for Gibbs distributions.
method Uses Markov chain Monte Carlo with conditions on the chain's spectral gap and Poincaré inequality.
result Achieves similar sample complexity as i.i.d. samples with better query complexity.
We prove that the spectral gap of a finite planar graph X is bounded by $λ_1(X)\le C(\frac{\log(\diam X)}{\diam X})^2$ where C depends only on the degree of X. We then give a sequence of such graphs showing the the above estimate cannot be improved. This yields a negative answer to a question of Benjamini and Cur…
The paper improves L2-estimates for Dirac-Dolbeault operators on complex manifolds.
problem Improving L2-estimates for Dirac-Dolbeault operators on complex manifolds. method Generalized classical method to handle mixed curvature cases and provided bounds on error terms.
result Full asymptotic expansion for Bergman kernel obtained.
Study spectral properties on manifolds with conical singularities, proving new inequalities.
problem Analyzing spectral properties and geometric inequalities on manifolds with conical singularities.
method Develops new inequalities for manifolds with conical singularities, not covered by existing methods.
result Proves a Bakry-Émery inequality, Hardy inequality, and spectral gap estimate.
Improved spectral gap for MwG with adaptive RWM proposals.
problem Improving mixing efficiency of MwG for log-concave distributions.
method Using adaptive RWM proposals tuned to match conditional variances of log-concave target distributions.
result Established a spectral gap lower bound of order O(1/κd) for MwG. A reverse Riesz estimate and spectral gap imply a Poincaré inequality.
problem Establishing a Poincaré inequality using a reverse Riesz estimate and spectral gap.
method Combining a reverse Riesz estimate and spectral gap condition to prove a Poincaré inequality.
result A Poincaré inequality is derived from a reverse Riesz estimate and spectral gap condition.
Study spectral gaps in hyperbolic rational homology spheres.
problem Finding spectral gaps in hyperbolic rational homology spheres.
method Construction of families of hyperbolic rational homology spheres with coexact 1-form spectral gaps.
result Provided intervals containing limit points of spectral gaps, with the rightmost interval being [0.8196, 0.8277].
Estimates intrinsic dimension of data sets robustly to noise.
problem Estimating intrinsic dimension of noisy data sets.
method Quantum Cognition Machine Learning for data representation and spectral gap detection.
result Robust estimation of intrinsic dimension in the presence of Gaussian noise.
We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In…
Sharp Hardy and spectral gap inequalities found on special irreversible Finsler manifolds.
problem Understanding Hardy and spectral gap inequalities on irreversible Finsler manifolds.
method Finslerian extension of the method of Riccati pairs.
result Sharpness of Hardy and spectral gap inequalities on specific Finsler manifolds.
For large genus, spectral gaps on hyperbolic surfaces approach a limit.
problem Understanding spectral gaps on hyperbolic surfaces of large genus.
method Analyzing the maximum of λk−λk−1 over thick parts of moduli spaces. result The maximum of λk−λk−1 approaches 41 for large genus. A new method simulates a lazy version of a Markov chain for empirical inference.
problem Estimating and testing unknown Markov chains with limited data.
method Simulates an α-lazy version of an unknown Markov chain, making it ergodic.
result The pseudo spectral gap can be applied to non-ergodic Markov chains.
Study shows spectral gaps limit points on surfaces.
problem Understanding spectral gaps on arithmetic hyperbolic surfaces.
method Analyzes closed arithmetic hyperbolic surfaces to find limit points of spectral gaps.
result Limit points of spectral gaps are on the interval [0, 1/4].
Constructs spin hyperbolic surfaces with a spectral gap for Dirac operator.
problem Finding spectral gaps for Dirac operators on hyperbolic surfaces.
method Explicit construction of spin hyperbolic surfaces with increasing genus.
result Uniform spectral gap for Dirac operator on constructed surfaces.
The paper explores how to increase Steklov spectral gaps on manifolds with fixed boundary.
problem Finding ways to increase Steklov spectral gaps on manifolds with fixed boundary.
method Constructing compact manifolds with fixed boundary geometry and applying localized conformal deformations.
result It is possible to make the spectral gap arbitrarily large using localized conformal deformations.
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.
Proves spectral gap for frame flows on hyperbolic manifolds.
problem Exponential mixing of frame flows on hyperbolic manifolds.
method Resolvent estimates and Borel-Weil calculus.
result Optimal essential spectral gap property for the generator.
Study shows optimal spectral gaps diminish in large genus surfaces.
problem Optimizing spectral gaps in large genus surfaces.
method Analysis of Weil-Petersson probability and eigenvalues of Laplacian.
result Probability of optimal spectral gaps vanishes as genus increases.
Confirms unique eigenfunction in hyperbolic packing has maximal spectral gap.
problem Sarnak's spectral gap question for hyperbolic packings.
method Analysis of Patterson-Sullivan base eigenfunctions and spectral gaps.
result Unique square-integrable eigenfunction has maximal spectral gap.
New theorem improves spectral gap for sampling from mixture distributions.
problem Sampling from multimodal distributions with simulated tempering.
method Introduced a decomposition theorem for the restricted spectral gap of simulated tempering.
result Lower bound on the restricted spectral gap for mixture distributions.
Study on spectral gaps of hyperbolic surfaces as genus increases.
problem Understanding differences in eigenvalues for large genus hyperbolic surfaces.
method Analysis of the Laplacian on degenerating hyperbolic surfaces, min-max principle.
result Supremum of spectral gaps has infimum limit of at least 1/4 as genus increases.
The study examines hyperbolic 3-manifolds with uniform spectral gaps for coclosed 1-forms.
problem Understanding the spectral gap for coclosed 1-forms in hyperbolic 3-manifolds.
method Constructing sequences of manifolds and analyzing their spectral properties and homology growth.
result Sequences of hyperbolic manifolds can have uniform spectral gaps for coclosed 1-forms but unbounded torsion homology growth.
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.
New expanders found using origami surfaces with spectral gap.
problem Constructing expanders with spectral gap on surfaces of arbitrary genus.
method Affine actions on origami surfaces to achieve spectral gap.
result New expanders distinct from classical ones.
Solves the Wiegold problem by showing free products of left-orderable groups have normal rank > 1.
problem Wiegold problem about groups of normal rank > 1
method Topological argument and intricate construction of left-orders
result Free products of nontrivial left-orderable groups have normal rank > 1
Existence proved for metrics maximizing eigenvalue on non-orientable surfaces.
problem Finding metrics maximizing the first eigenvalue on non-orientable surfaces.
method Proved existence under spectral gap conditions.
result Existence of metrics maximizing eigenvalue on non-orientable surfaces.
Researchers prove rigidity for spectral gap on special metric spaces.
problem Proving rigidity for spectral gap on RCD(K,∞)-spaces. method Lift of eigenfunctions to Wasserstein space, theory of regular Lagrangian flows.
result Sharp spectral gap achieved only by splitting off a 1-dimensional Gaussian space.
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 eigenvalue of Hodge Laplacian on coexact 1-forms in hyperbolic 3-manifolds is related to isoperimetric ratio.
problem Eigenvalue of Hodge Laplacian on coexact 1-forms in hyperbolic 3-manifolds.
method Using isoperimetric ratio relating geodesic length and stable commutator length, with comparison constants polynomial in volume and injectivity radius.
result Estimates show spectral gap of 1-form Laplacian vanishing exponentially fast in volume for certain hyperbolic 3-manifolds.
Spectral gap theorem for free products of groups.
problem Understanding the spectral gap of elements in free products of groups.
method Analyzing the structure of free products and properties of elements in their commutator subgroup.
result The spectral gap of a non-conjugate element in a free product of groups is at least 1/2.
Researchers prove a spectral gap for Hecke covers of Schottky surfaces.
problem Proving a spectral gap for Hecke congruence covers of arithmetic Schottky surfaces.
method Using the generalized Riemann hypothesis for quadratic L-functions and properties of Schottky subgroups.
result Established a uniform and explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces.
Sharp upper bounds found for Steklov eigenvalues of warped products.
problem Finding bounds for Steklov eigenvalues of specific metric configurations.
method Investigation of Steklov spectrum for warped products with a fiber of dimension 2.
result Sharp upper bounds for Steklov eigenvalues in terms of the eigenvalues of the Laplacian on the fiber.