Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the conjectural γ-Liouville quantum gravity (LQG) metric, obtained by exponentiating the discrete Gaussian free field (GFF) and the circle average regularization of the continuum GFF respectively. We show that these two models can be coupled so that with high probability distances in these models agree up to o(1) errors in the exponent, and thus have the same distance exponent.Ding and Gwynne (2018) give a formula for the continuum LFPP distance exponent in terms of the γ-LQG dimension exponent d γ . Using results of Ding and Li (2018) on the level set percolation of the discrete GFF, we bound the DLFPP distance exponent and hence obtain a new lower bound d γ ≥ 2 + γ 2 2 . This improves on previous lower bounds for d γ for the regime γ ∈ (γ 0 , 0.576), for some small nonexplicit γ 0 > 0.

Consider a critical (γ = 2) Liouville quantum gravity (LQG) disk together with an independent conformal loop ensemble (CLE) with parameter κ = 4. We show that the critical LQG surfaces parametrized by the regions enclosed by the CLE 4 loops are conditionally independent critical LQG disks given the LQG lengths of the loops. We also show that the joint law of the LQG lengths of the loops is described in terms of the jumps of a certain 3/2-stable process. Our proofs are via a limiting argument based on the analogous statements for γ ∈ ( 8/3, 2) and κ = γ 2 ∈ (8/3, 4) which were proven by Miller, Sheffield, and Werner (2020). Our results are used in the construction of a coupling of supercritical LQG with CLE 4 in another paper by the same authors.

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We demonstrate that the conformal loop ensemble (CLE) has a rich integrable structure by establishing exact formulas for two CLE observables. The first describes the joint moments of the conformal radii of loops surrounding three points for CLE on the sphere. Up to normalization, our formula agrees with the imaginary DOZZ formula due to Zamolodchikov ( 2005) and Kostov-Petkova ( 2007), which is the three-point structure constant of certain conformal field theories that generalize the minimal models. This verifies the CLE interpretation of the imaginary DOZZ formula by Ikhlef, Jacobsen and Saleur (2015). Our second result is for the moments of the electrical thickness of CLE loops first considered by Kenyon and Wilson (2004). Our proofs rely on the conformal welding of random surfaces and two sources of integrability concerning CLE and Liouville quantum gravity (LQG). First, LQG surfaces decorated with CLE inherit a rich integrable structure from random planar maps decorated with the O(n) loop model. Second, as the field theory describing LQG, Liouville conformal field theory is integrable. In particular, the DOZZ formula and the FZZ formula for its structure constants are crucial inputs to our results.

We derive the large deviation principle for radial Schramm-Loewner evolution (SLE) on the unit disk with parameter κ → ∞. Restricting to the time interval [0, 1], the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures {φ 2 t (ζ) dζ} t∈[0,1] on the unit circle andOur proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.