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All 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.

Critical Liouville measure as a limit of subcritical measures (2019)
Journal Article
Aru, J., Powell, E., & Sepúlveda, A. (2019). Critical Liouville measure as a limit of subcritical measures. Electronic Communications in Probability, 24, 1-16. https://doi.org/10.1214/19-ecp209

We study how the Gaussian multiplicative chaos (GMC) measures μγ corresponding to the 2D Gaussian free field change when γ approaches the critical parameter 2. In particular, we show that as γ→2−, (2−γ)−1μγ converges in probability to 2μ′, where μ′ i... Read More about Critical Liouville measure as a limit of subcritical measures.

An invariance principle for branching diffusions in bounded domains (2018)
Journal Article
Powell, E. (2019). An invariance principle for branching diffusions in bounded domains. Probability Theory and Related Fields, 173(3-4), 999-1062. https://doi.org/10.1007/s00440-018-0847-8

We study branching diffusions in a bounded domain D of Rd in which particles are killed upon hitting the boundary ∂D . It is known that any such process undergoes a phase transition when the branching rate β exceeds a critical value: a multiple of th... Read More about An invariance principle for branching diffusions in bounded domains.

Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation (2018)
Journal Article
Powell, E. (2018). Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation. Electronic Journal of Probability, 23, 1-26. https://doi.org/10.1214/18-ejp157

We show that, for general convolution approximations to a large class of log-correlated fields, including the 2d Gaussian free field, the critical chaos measures with derivative normalisation converge to a limiting measure µ This limiting measure doe... Read More about Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation.

Level lines of the Gaussian free field with general boundary data (2017)
Journal Article
Powell, E., & Wu, H. (2017). Level lines of the Gaussian free field with general boundary data. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 53(4), 2229-2259. https://doi.org/10.1214/16-aihp789

We study the level lines of a Gaussian free field in a planar domain with general boundary data F. We show that the level lines exist as continuous curves under the assumption that F is regulated (i.e., admits finite left and right limits at every po... Read More about Level lines of the Gaussian free field with general boundary data.