School of Mathematics and Statistics - Research Publications
Permanent URI for this collection
Search Results
1 - 10 of 14
-
ItemNo Preview AvailableAsymptotics for the critical level and a strong invariance principle for high intensity shot noise fields(Institute of Mathematical Statistics, 2023-01-01)
-
ItemBoundedness of the nodal domains of additive Gaussian fields(American Mathematical Society, 2022)We study the connectivity of the excursion sets of additive Gaussian fields, i.e. stationary centred Gaussian fields whose covariance function decomposes into a sum of terms that depend separately on the coordinates. Our main result is that, under mild smoothness and correlation decay assumptions, the excursion sets {�≤ℓ} of additive planar Gaussian fields are bounded almost surely at the critical level ℓ�=0. Since we do not assume positive correlations, this provides the first examples of continuous non-positively-correlated stationary planar Gaussian fields for which the boundedness of the nodal domains has been confirmed. By contrast, in dimension �≥3 the excursion sets have unbounded components at all levels.
-
ItemSharp phase transition for Cox percolation(INST MATHEMATICAL STATISTICS-IMS, 2022)We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence, satisfies a local boundedness condition and can be constructed from a discrete iid random field, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction that allows us to compare different notions of influence.
-
ItemUpper bounds on the one-arm exponent for dependent percolation models(SPRINGER HEIDELBERG, 2023-02)We prove upper bounds on the one-arm exponent η1 for a class of dependent percolation models which generalise Bernoulli percolation; while our main interest is level set percolation of Gaussian fields, the arguments apply to other models in the Bernoulli percolation universality class, including Poisson–Voronoi and Poisson–Boolean percolation. More precisely, in dimension d= 2 we prove that η1≤ 1 / 3 for continuous Gaussian fields with rapid correlation decay (e.g. the Bargmann–Fock field), and in d≥ 3 we prove η1≤ d/ 3 for finite-range fields, both discrete and continuous, and η1≤ d- 2 for fields with rapid correlation decay. Although these results are classical for Bernoulli percolation (indeed they are best-known in general), existing proofs do not extend to dependent percolation models, and we develop a new approach based on exploration and relative entropy arguments. The proof also makes use of a new Russo-type inequality for Gaussian fields, which we apply to prove the sharpness of the phase transition and the mean-field bound for finite-range fields.
-
ItemAsymptotics for the critical level and a strong invariance principle for high intensity shot noise fields( 2021-11-17)We study _ne properties of the convergence of a high intensity shot noise _eld towards the Gaussian _eld with the same covariance structure. In particular we (i) establish a strong invariance principle, i.e. a quantitative coupling between a high intensity shot noise _eld and the Gaussian limit such that they are uniformly close on large domains with high probability, and (ii) use this to derive an asymptotic expansion for the critical level above which the excursion sets of the shot noise _eld percolate.
-
ItemNo Preview AvailableFluctuations of the number of excursion sets of planar Gaussian fields(Mathematical Sciences Publishers, 2022-05-11)For a smooth, stationary, planar Gaussian field, we consider the number of connected components of its excursion set (or level set) contained in a large square of area R2. The mean number of components is known to be of order R2 for generic fields and all levels. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain levels ℓ, these random variables have fluctuations of order at least R, and hence variance of order at least R2. In particular this holds for excursion sets when ℓ is in some neighbourhood of zero, and it holds for excursion/level sets when ℓ is sufficiently large. We prove stronger fluctuation lower bounds of order Rα for α ∈ [1, 2] in the case that the spectral density has a singularity at the origin. Finally we show that the number of excursion/level sets for the random plane wave at certain levels has fluctuations of order at least R3/2, and hence variance of order at least R3. We expect that these bounds are of the correct order, at least for generic levels.
-
ItemPercolation of the excursion sets of planar symmetric shot noise fields(Elsevier, 2022-05)We prove the existence of phase transitions in the global connectivity of the excursion sets of planar symmetric shot noise fields. Our main result establishes a phase transition with respect to the level for shot noise fields with symmetric log-concave mark distributions, including Gaussian, uniform, and Laplace marks, and kernels that are positive, symmetric, and have sufficient tail decay. Without the log-concavity assumption we prove a phase transition with respect to the intensity of positive marks.
-
ItemFluctuations of the number of excursion sets of planar Gaussian fields( 2019-08-28)For a smooth, stationary, planar Gaussian field, we consider the number of connected components of its excursion set (or level set) contained in a large square of area R2. The mean number of components is known to be of order R2 for generic fields and all levels. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain levels ℓ, these random variables have fluctuations of order at least R, and hence variance of order at least R2. In particular this holds for excursion sets when ℓ is in some neighbourhood of zero, and it holds for excursion/level sets when ℓ is sufficiently large. We prove stronger fluctuation lower bounds of order Rα, α ∈ [1, 2], in the case that the spectral density has a singularity at the origin. Finally we show that the number of excursion/level sets for the Random Plane Wave at certain levels has fluctuations of order at least R3/2, and hence variance of order at least R3. We expect that these bounds are of the correct order, at least for generic levels.
-
ItemNo Preview AvailableA COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS(INST MATHEMATICAL STATISTICS, 2020-11)
-
ItemNo Preview AvailableSmoothness and monotonicity of the excursion set density of planar gaussian fields(Institute of Mathematical Statistics, 2020-01-01)Nazarov and Sodin have shown that the number of connected components of the nodal set of a planar Gaussian field in a ball of radius R, normalised by area, converges to a constant as R → ∞. This has been generalised to excursion/level sets at arbitrary levels, implying the existence of functionals cES(ℓ) and cLS(ℓ) that encode the density of excursion/level set components at the level ℓ. We prove that these functionals are continuously differentiable for a wide class of fields. This follows from a more general result, which derives differentiability of the functionals from the decay of the probability of ‘four-arm events’ for the field conditioned to have a saddle point at the origin. For some fields, including the important special cases of the Random Plane Wave and the Bargmann-Fock field, we also derive stochastic monotonicity of the conditioned field, which allows us to deduce regions on which cES(ℓ) and cLS(ℓ) are monotone.