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2020

Reconstructing the base field from imaginary multiplicative chaos

Aru, Juhan, Junnila, Janne

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We show that the imaginary multiplicative chaos $\exp(i\beta \Gamma)$ determines the gradient of the underlying field $\Gamma$ for all log-correlated Gaussian fields with covariance of the form $-\log |x-y| + g(x,y)$ with mild regularity conditions on $g$, for all $d \geq 2$ and for all $\beta \in (0,\sqrt{d})$.

In particular, we show that the 2D continuum zero boundary Gaussian free field is measurable w.r.t.
its imaginary chaos.

Reconstructing the base field from imaginary multiplicative chaos · Around

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H. Lacoin, R. Rhodes, V. Vargas: A probabilistic approach of ultraviolet renormalisation in the boundary Sine-Gordon model. arXiv preprint (2019), arXiv:1903.01394
Original
1903
Earlier work this paper cites.
L. Schoug, A. Sepúlveda, and F. Viklund: Dimension of two-valued sets via imaginary chaos. arXiv preprint (2019), arXiv:1910.09294
Original
1910
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J. Kahane: Sur le chaos multiplicatif. Ann. Sci. Math. Québec, (1985), Volume 9, Issue 2
1985
Earlier work this paper cites.
C. Garban and A. Sepúlveda: Statistical reconstruction of the Gaussian free field and KT transition. arXiv preprint (2020), arXiv:2002.12284
Original
2002
Earlier work this paper cites.

Similar

E. Powell and W. Werner: Lecture notes on the Gaussian Free Field. arXiv preprint (2020), arXiv:2004.04720
Original
2004
Cited alongside, same era.
N. Berestycki, S. Sheffield and X. Sun: Equivalence of Liouville measure and Gaussian free field. arXiv preprint (2014), arXiv:1410.5407
Original
2014
Cited alongside, same era.
R. Rhodes and V. Vargas: Gaussian multiplicative chaos and applications: A review. Probab. Surveys (2014), Volume 11
2014
Cited alongside, same era.
N. Berestycki, Introduction to the Gaussian Free Field and Liouville Quantum Gravity. Lecture notes (2016), available on authors webpage
2016
Cited alongside, same era.

Then

J. Junnila, E. Saksman and C. Webb: Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model. arXiv preprint (2018), arXiv:1806.02118. To appear in Annals of Applied Probability
Original
2018
Later among the works it cites.
J. Aru: Gaussian Multiplicative Chaos Through the Lens of the 2D Gaussian Free Field. Markov Process. Relat. Fields (2020), Volume 26, Issue 1
2020
Closest in time.
J. Aru, E. Powell and A. Sepúlveda: Liouville measure as a multiplicative cascade via the level sets of the Gaussian free field. Ann. Inst. Fourier (2020), Volume 70, Issue 1
2020
Closest in time.

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