Journal: Geometric and Functional Analysis

Abbreviation

Geom. Funct. Anal.

Publisher

Springer

Journal Volumes

ISSN

1016-443X
1420-8970

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Publications 1 - 10 of 13
  • Sharpness and Locality for Percolation on Finite Transitive Graphs
    Item type: Journal Article
    Easo, Philip (2025)
    Geometric and Functional Analysis
    Let (Gn)=((Vn,En)) be a sequence of finite connected vertex-transitive graphs with uniformly bounded vertex degrees such that |Vn|→∞ as n→∞. We say that percolation on Gn has a sharp phase transition (as n→∞) if, as the percolation parameter crosses some critical point, the number of vertices contained in the largest percolation cluster jumps from logarithmic to linear order with high probability. We prove that percolation on Gn has a sharp phase transition unless, after passing to a subsequence, the rescaled graph-metric on Gn (rapidly) converges to the unit circle with respect to the Gromov-Hausdorff metric. We deduce that under the same hypothesis, the critical point for the emergence of a giant (i.e. linear-sized) cluster in Gn coincides with the critical point for the emergence of an infinite cluster in the Benjamini-Schramm limit of (Gn), when this limit exists.
  • A proof of Ringel’s conjecture
    Item type: Journal Article
    Montgomery, Richard; Pokrovskiy, Alexey; Sudakov, Benny (2021)
    Geometric and Functional Analysis
    A typical decomposition question asks whether the edges of some graph G can be partitioned into disjoint copies of another graph H. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with n edges packs 2n + 1 times into the complete graph K2n+1. In this paper, we prove this conjecture for large n.
  • A logarithmic epiperimetric inequality for the obstacle problem
    Item type: Journal Article
    Colombo, Maria; Spolaor, Luca; Velichkov, Bozhidar (2018)
    Geometric and Functional Analysis
  • Symmetry results for critical anisotropic p-Laplacian equations in convex cones
    Item type: Journal Article
    Ciraolo, Giulio; Figalli, Alessio; Roncoroni, Alberto (2020)
    Geometric and Functional Analysis
    Given n ≥ 2 and 1
  • An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture
    Item type: Journal Article
    Chen, Yuansi (2021)
    Geometric and Functional Analysis
    We prove an almost constant lower bound of the isoperimetric coefficient in the KLS conjecture. The lower bound has the dimension dependency d-od(1). When the dimension is large enough, our lower bound is tighter than the previous best bound which has the dimension dependency d . Improving the current best lower bound of the isoperimetric coefficient in the KLS conjecture has many implications, including improvements of the current best bounds in Bourgain’s slicing conjecture and in the thin-shell conjecture, better concentration inequalities for Lipschitz functions of log-concave measures and better mixing time bounds for MCMC sampling algorithms on log-concave measures. - 1 / 4
  • Coalescence of Geodesics and the BKS Midpoint Problem in Planar First-Passage Percolation
    Item type: Journal Article
    Dembin, Barbara; Elboim, Dor; Peled, Ron (2024)
    Geometric and Functional Analysis
    We consider first-passage percolation on Z² with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has more than 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics. The result leads to a quantitative resolution of the Benjamini–Kalai–Schramm midpoint problem. It is shown that the probability that the geodesic between two given points passes through a given edge is smaller than a power of the distance between the points and the edge. We further prove that the limit shape assumption is satisfied for a specific family of distributions. Lastly, related to the 1965 Hammersley–Welsh highways and byways problem, we prove that the expected fraction of the square {−n,…,n}2 which is covered by infinite geodesics starting at the origin is at most an inverse power of n. This result is obtained without explicit limit shape assumptions.
  • Algebraic twists of modular forms and Hecke orbits
    Item type: Journal Article
    Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe (2015)
    Geometric and Functional Analysis
    We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the ℓ-adic Fourier transform introduced by Deligne and studied by Katz and Laumon.
  • Isoperimetric inequalities of Euclidean type in metric spaces
    Item type: Journal Article
    Wenger, S. (2005)
    Geometric and Functional Analysis
  • Maximal Multiplicity of Laplacian Eigenvalues in Negatively Curved Surfaces
    Item type: Journal Article
    Letrouit, Cyril; Machado, Simon (2024)
    Geometric and Functional Analysis
    In this work, we obtain the first upper bound on the multiplicity of Laplacian eigenvalues for negatively curved surfaces which is sublinear in the genus g. Our proof relies on a trace argument for the heat kernel, and on the idea of leveraging an r-net in the surface to control this trace. This last idea was introduced in 2021 for similar spectral purposes in the context of graphs of bounded degree. Our method is robust enough to also yield an upper bound on the “approximate multiplicity” of eigenvalues, i.e., the number of eigenvalues in windows of size 1/logβ(g), β>0. This work provides new insights on a conjecture by Colin de Verdière and new ways to transfer spectral results from graphs to surfaces.
  • Burgess-like Subconvex Bounds for GL2 × GL1
    Item type: Journal Article
    Wu, Han (2014)
    Geometric and Functional Analysis
    Let F be a number field, π an irreducible cuspidal representation of GL2(AF) with unitary central character, and χ a Hecke character of analytic conductor Q. Then L(1/2,π⊗χ)≪Q12−18(1−2θ)+ϵ, where 0≤θ≤1/2 is any exponent towards the Ramanujan–Petersson conjecture. The proof is based on an idea of unipotent translation originated from P. Sarnak then developed by Ph. Michel and A. Venkatesh, combined with a method of amplification.
Publications 1 - 10 of 13