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Outputs (14)

Many-to-few for non-local branching Markov process (2024)
Journal Article
Harris, S. C., Horton, E., Kyprianou, A. E., & Powell, E. (2024). Many-to-few for non-local branching Markov process. Electronic Journal of Probability, 29, 1-26. https://doi.org/10.1214/24-ejp1098

We provide a many-to-few formula in the general setting of non-local branching Markov processes. This formula allows one to compute expectations of k-fold sums over functions of the population at k different times. The result generalises [13] to the... Read More about Many-to-few for non-local branching Markov process.

Thick points of the planar GFF are totally disconnected for all γ≠0 (2023)
Journal Article
Aru, J., Papon, L., & Powell, E. (2023). Thick points of the planar GFF are totally disconnected for all γ≠0. Electronic Journal of Probability, 28, 1-24. https://doi.org/10.1214/23-ejp975

We prove that the set of γ-thick points of a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all γ≠0. Our proof relies on the coupling between a GFF and the nested
CLE4. In particular, we show th... Read More about Thick points of the planar GFF are totally disconnected for all γ≠0.

Brownian half‐plane excursion and critical Liouville quantum gravity (2022)
Journal Article
Aru, J., Holden, N., Powell, E., & Sun, X. (2023). Brownian half‐plane excursion and critical Liouville quantum gravity. Journal of the London Mathematical Society, 107(1), 441-509. https://doi.org/10.1112/jlms.12689

In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm–Loewner evolutions (SLE) can be obtained by gluing together a pair of Brownian motions. In this paper, we study th... Read More about Brownian half‐plane excursion and critical Liouville quantum gravity.

A characterisation of the continuum Gaussian free field in arbitrary dimensions (2022)
Journal Article
Aru, J., & Powell, E. (2022). A characterisation of the continuum Gaussian free field in arbitrary dimensions. Journal de l’École polytechnique — Mathématiques, 9, 1101-1120. https://doi.org/10.5802/jep.201

e prove that under certain mild moment and continuity assumptions, the d-dimensional continuum Gaussian free field is the only stochastic process satisfying the usual domain Markov property and a scaling assumption. Our proof is based on a decomposit... Read More about A characterisation of the continuum Gaussian free field in arbitrary dimensions.

Critical Gaussian multiplicative chaos: a review (2021)
Journal Article
Powell, E. (2021). Critical Gaussian multiplicative chaos: a review. Markov processes and related fields, 27(4), 557-606

This review-style article presents an overview of recent progress in constructing and studying critical Gaussian multiplicative chaos. A proof that the critical measure in any dimension can be obtained as a limit of subcritical measures is given.

Conformal welding for critical Liouville quantum gravity (2021)
Journal Article
Holden, N., & Powell, E. (2021). Conformal welding for critical Liouville quantum gravity. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 57(3), 1229-1254. https://doi.org/10.1214/20-aihp1116

Consider two critical Liouville quantum gravity surfaces (i.e., γ-LQG for γ = 2), each with the topology of H and with infinite boundary length. We prove that there a.s. exists a conformal welding of the two surfaces, when the boundaries are identifi... Read More about Conformal welding for critical Liouville quantum gravity.

(1+𝜀) moments suffice to characterise the GFF (2021)
Journal Article
Berestycki, N., Powell, E., & Ray, G. (2021). (1+𝜀) moments suffice to characterise the GFF. Electronic Journal of Probability, 26(44), 1-25. https://doi.org/10.1214/20-ejp566

We show that there is “no stable free field of index α ∈ ( 1 , 2 ) ”, in the following sense. It was proved in [4] that subject to a fourth moment assumption, any random generalised function on a domain D of the plane, satisfying conformal invariance... Read More about (1+𝜀) moments suffice to characterise the GFF.

Liouville measure as a multiplicative cascade via level sets of the Gaussian free field (2020)
Journal Article
Aru, J., Powell, E., & Sepúlveda, A. (2020). Liouville measure as a multiplicative cascade via level sets of the Gaussian free field. Annales de l'Institut Fourier, 70(1), 245-205. https://doi.org/10.5802/aif.3312

We provide new constructions of the subcritical and critical Gaussian multiplicative chaos (GMC) measures corresponding to the 2D Gaussian free field (GFF). As a special case we recover E. Aidekon’s construction of random measures using nested confor... Read More about Liouville measure as a multiplicative cascade via level sets of the Gaussian free field.

A characterisation of the Gaussian free field (2019)
Journal Article
Berestycki, N., Powell, E., & Ray, G. (2020). A characterisation of the Gaussian free field. Probability Theory and Related Fields, 176(3-4), 1259-1301. https://doi.org/10.1007/s00440-019-00939-9

We prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption.