The giant component of excursion sets of spherical Gaussian ensembles: existence, uniqueness, and volume concentration

TY - JOUR

T1 - The giant component of excursion sets of spherical Gaussian ensembles: existence, uniqueness, and volume concentration

AU - Wigman, Igor

AU - Muirhead, Stephen

PY - 2025/2/19

Y1 - 2025/2/19

N2 - We establish the existence and uniqueness of a well-concentrated giant component in the supercritical excursion sets of three important ensembles of spherical Gaussian random fields: Kostlan's ensemble, band-limited ensembles, and the random spherical harmonics. Our main results prescribe quantitative bounds for the volume fluctuations of the giant that are essentially optimal for non-monochromatic ensembles, and suboptimal but still strong for monochromatic ensembles.Our results support the emerging picture that giant components in Gaussian random field excursion sets have similar large-scale statistical properties to giant components in supercritical Bernoulli percolation. The proofs employ novel decoupling inequalities for spherical ensembles which are of independent interest.

AB - We establish the existence and uniqueness of a well-concentrated giant component in the supercritical excursion sets of three important ensembles of spherical Gaussian random fields: Kostlan's ensemble, band-limited ensembles, and the random spherical harmonics. Our main results prescribe quantitative bounds for the volume fluctuations of the giant that are essentially optimal for non-monochromatic ensembles, and suboptimal but still strong for monochromatic ensembles.Our results support the emerging picture that giant components in Gaussian random field excursion sets have similar large-scale statistical properties to giant components in supercritical Bernoulli percolation. The proofs employ novel decoupling inequalities for spherical ensembles which are of independent interest.

M3 - Article

SN - 0178-8051

JO - PROBABILITY THEORY AND RELATED FIELDS

JF - PROBABILITY THEORY AND RELATED FIELDS

ER -