The distribution of Gaussian multiplicative chaos on the unit interval

Rémy, Guillaume; Zhu, Tunan

doi:10.1214/19-aop1377

Cited by 18 publications

(27 citation statements)

References 44 publications

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“…However, there is nothing obviously special about γ = √ 8/3 from either of the definitions of the LQG metric given in this paper (the limit of LFPP or the axiomatic definition). There has been a recent proliferation of exact formulas for quantities related to the γ -LQG area and boundary length measures for general γ ∈ (0, 2), proven using ideas from conformal field theory: see, e.g., [54,75,77]. In the special case when γ = √ 8/3, exact formulas for various quantities associated with the √ 8/3-LQG metric can be obtained using its connection to the Brownian surfaces.…”

Section: Additional Properties Of the Lqg Metricmentioning

confidence: 99%

Existence and uniqueness of the Liouville quantum gravity metric for $$\gamma \in (0,2)$$

2020

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We show that for each $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , there is a unique metric (i.e., distance function) associated with $$\gamma $$ γ -Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) h, there is a unique random metric $$D_h$$ D h associated with the Riemannian metric tensor “$$e^{\gamma h} (dx^2 + dy^2)$$ e γ h ( d x 2 + d y 2 ) ” on $${\mathbb {C}}$$ C which is characterized by a certain list of axioms: it is locally determined by h and it transforms appropriately when either adding a continuous function to h or applying a conformal automorphism of $$\mathbb {C}$$ C (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The $$\gamma $$ γ -LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding et al. (Tightness of Liouville first passage percolation for $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , 2019. arXiv:1904.08021) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , our metric coincides with the $$\sqrt{8/3}$$ 8 / 3 -LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , we conjecture that our metric is the Gromov–Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems.

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Section: Additional Properties Of the Lqg Metricmentioning

confidence: 99%

Existence and uniqueness of the Liouville quantum gravity metric for $$\gamma \in (0,2)$$

2020

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“…In the discrete setting, they compute the first few low moments (which agree with the first few explicit examples computed in Appendix A) and establish a connection to the KPP equation in the continuous setting. Additionally, it is natural to ask if the leading order coefficient ρ(k, β) for kβ 2 < 1 in (14) could take the form f (β) k Γ (1 − kβ 2 ) for some function f , in line with Fyodorov-Bouchaud [17] or Remy-Zhu [28] formulae for related problems in the same regime (see also (13)). Such a statement does not appear to hold here.…”

Section: Resultsmentioning

confidence: 96%

“…For example, ρ(2, β), σ(2, β), and τ (2, β) are given respectively by (27),(28), and (30). Although the k = 1 case exhibits no phase transition, we will write for ease of notationρ(1, β) ≡ σ (1, β) ≡ τ (1, β) = 1.…”

mentioning

confidence: 99%

Moments of Moments and Branching Random Walks

Bailey

Keating

2021

J Stat Phys

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We calculate, for a branching random walk $$X_n(l)$$ X n ( l ) to a leaf l at depth n on a binary tree, the positive integer moments of the random variable $$\frac{1}{2^{n}}\sum _{l=1}^{2^n}e^{2\beta X_n(l)}$$ 1 2 n ∑ l = 1 2 n e 2 β X n ( l ) , for $$\beta \in {\mathbb {R}}$$ β ∈ R . We obtain explicit formulae for the first few moments for finite n. In the limit $$n\rightarrow \infty $$ n → ∞ , our expression coincides with recent conjectures and results concerning the moments of moments of characteristic polynomials of random unitary matrices, supporting the idea that these two problems, which both fall into the class of logarithmically correlated Gaussian random fields, are related to each other.

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“…The BPZ equations are essential for proving integrability of LCFT. They were used in the proof of the DOZZ-formula [14,15] for the 3-point function of LCFT on the sphere, and after this similar methods were used for obtaining integrability results for one dimensional GMC measures on the unit circle [19] and on the unit interval [20]. The unit circle computation was based on a boundary LCFT, which is defined in [11].…”

Section: 2mentioning

confidence: 99%