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1. Permutons, meanders, and SLE-decorated Liouville quantum gravity

   https://doi.org/10.4171/jems/1646  

   Borga, Jacopo; Gwynne, Ewain; Sun, Xin (June 2025, Journal of the European Mathematical Society)

   We study a class of random permutons which can be constructed from a pair of space-filling Schramm–Loewner evolution (SLE) curves on a Liouville quantum gravity (LQG) surface. This class includes the skew Brownian permutons introduced by Borga (2021), which describe the scaling limit of various types of random pattern-avoiding permutations. Another interesting permuton in our class is the meandric permuton, which corresponds to two independent\mathrm{SLE}_{8}curves on a \gamma-LQG surface with\gamma = \sqrt{(17-\sqrt{\vphantom{a}145})/3}. Building on work by Di Francesco, Golinelli, and Guitter (2000), we conjecture that the meandric permuton describes the scaling limit of uniform meandric permutations, i.e., the permutations induced by a simple loop in the plane which crosses a line a specified number of times. We show that for any sequence of random permutations which converges to one of the above random permutons, the length of the longest increasing subsequence is sublinear. This proves that the length of the longest increasing subsequence is sublinear for Baxter, strong-Baxter, and semi-Baxter permutations and leads to the conjecture that the same is true for meandric permutations. We also prove that the closed support of each of the random permutons in our class has Hausdorff dimension one. Finally, we prove a re-rooting invariance property for the meandric permuton and write down a formula for its expected pattern densities in terms of LQG correlation functions (which are known explicitly) and the probability that an\mathrm{SLE}_{8}hits a given set of points in numerical order (which is not known explicitly). We conclude with a list of open problems. 

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2. Gaussian curvature on random planar maps and Liouville quantum gravity

   https://doi.org/10.1007/s00440-025-01430-4  

   Hip, Andres_A Contreras; Gwynne, Ewain (April 2026, Probability Theory and Related Fields)

   Abstract We investigate the notion of curvature in the context of Liouville quantum gravity (LQG) surfaces. We define the Gaussian curvature for LQG, which we conjecture is the scaling limit of discrete curvature on random planar maps. Motivated by this, we study asymptotics for the discrete curvature of$$\epsilon $$ $ϵ$ -mated CRT maps. More precisely, we prove that the discrete curvature integrated against a$$C_c^2$$ $C_c^2$ test function is of order$$\epsilon ^{o(1)},$$ ${ϵ}^{o\left(1\right)},$ which is consistent with our scaling limit conjecture. On the other hand, we prove the total discrete curvature on a fixed space-filling SLE segment scaled by$$\epsilon ^{\frac{1}{4}}$$ ${ϵ}^{\frac{1}{4}}$ converges in distribution to an explicit random variable. 

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3. Harmonic balls in Liouville quantum gravity

   https://doi.org/10.1112/plms.70018  

   Bou‐Rabee, Ahmed; Gwynne, Ewain (January 2025, Proceedings of the London Mathematical Society)

   Abstract Harmonic balls are domains that satisfy the mean‐value property for harmonic functions. We establish the existence and uniqueness of harmonic balls on Liouville quantum gravity (LQG) surfaces using the obstacle problem formulation of Hele–Shaw flow. We show that LQG harmonic balls are neither Lipschitz domains nor LQG metric balls, and that the boundaries of their complementary connected components are Jordan curves. We conjecture that LQG harmonic balls are the scaling limit of internal diffusion limited aggregation on random planar maps. In a companion paper, we prove this in the special case of mated‐CRT maps. 

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4. Level repulsion in $$ \mathcal{N} $$ = 4 super-Yang-Mills via integrability, holography, and the bootstrap

   https://doi.org/10.1007/JHEP07(2024)059  

   Chester, Shai M; Dempsey, Ross; Pufu, Silviu S (July 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We combine supersymmetric localization with the numerical conformal bootstrap to bound the scaling dimension and OPE coefficient of the lowest-dimension unprotected operator in$$ \mathcal{N} $$ $N$ = 4 SU(N) super-Yang-Mills theory for a wide range ofNand Yang-Mills couplingsg_YM. We find that our bounds are approximately saturated by weak coupling results at smallg_YM. Furthermore, at largeNour bounds interpolate between integrability results for the Konishi operator at smallg_YMand strong-coupling results, including the first few stringy corrections, for the lowest-dimension double-trace operator at largeg_YM. In particular, our scaling dimension bounds describe the level splitting between the single- and double-trace operators at intermediate coupling. 

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5. Weak LQG metrics and Liouville first passage percolation

   https://doi.org/10.1007/s00440-020-00979-6  

   Dubédat, Julien; Falconet, Hugo; Gwynne, Ewain; Pfeffer, Joshua; Sun, Xin (October 2020, Probability Theory and Related Fields)

   null (Ed.)

   Abstract For $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , we define a weak $$\gamma $$ γ - Liouville quantum gravity ( LQG ) metric to be a function $$h\mapsto D_h$$ h ↦ D h which takes in an instance of the planar Gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms. We show that these axioms are satisfied for any subsequential limits of Liouville first passage percolation. Such subsequential limits were proven to exist by Ding et al. (Tightness of Liouville first passage percolation for $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , 2019. ArXiv e-prints, arXiv:1904.08021 ). It is also known that these axioms are satisfied for the $$\sqrt{8/3}$$ 8 / 3 -LQG metric constructed by Miller and Sheffield (2013–2016). For any weak $$\gamma $$ γ -LQG metric, we obtain moment bounds for diameters of sets as well as point-to-point, set-to-set, and point-to-set distances. We also show that any such metric is locally bi-Hölder continuous with respect to the Euclidean metric and compute the optimal Hölder exponents in both directions. Finally, we show that LQG geodesics cannot spend a long time near a straight line or the boundary of a metric ball. These results are used in subsequent work by Gwynne and Miller which proves that the weak $$\gamma $$ γ -LQG metric is unique for each $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , which in turn gives the uniqueness of the subsequential limit of Liouville first passage percolation. However, most of our results are new even in the special case when $$\gamma =\sqrt{8/3}$$ γ = 8 / 3 . 

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