We derive the alternating arm exponents of the critical Ising model. We obtain six different patterns of alternating boundary arm exponents which correspond to the boundary conditions (⊖⊕), (⊖ free) and (free free), and the alternating interior arm exponents.

We prove that uniform random quadrangulations of the sphere with$n$faces, endowed with the usual graph distance and renormalized by$n$^{−1/4}, converge as$n$→$∞$in distribution for the Gromov–Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called$Brownian map$, which was introduced by Marckert–Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of$geodesic stars$in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.

The tip multifractal spectrum of a 2-dimensional curve is one way to describe the behavior of the uniformizing conformal map of the complement near the tip. We give the tip multifractal spectrum for a Schramm–Loewner evolution (SLE) curve, we prove that the spectrum is valid with probability 1, and we give applications to the scaling of harmonic measure at the tip.

We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $langle v, (X^* X - z)^{-1} w rangle - langle v,w rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v, w in mathbb C^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $Im z geq N^{-1+epsilon}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.

We introduce the dynamical sine-Gordon equation in two space dimensions with parameter β, which is the natural dynamic associated to the usual quantum sine- Gordon model. It is shown that when β2 ∈ (0, 16π ) the Wick renormalised equation 3 is well-posed. In the regime β2 ∈ (0,4π), the Da Prato–Debussche method [J Funct Anal 196(1):180–210, 2002; Ann Probab 31(4):1900–1916, 2003] applies, while for β2 ∈ [4π, 16π ), the solution theory is provided via the theory of regularity structures 3 [Hairer, Invent Math 198(2):269–504, 2014]. We also show that this model arises naturally from a class of 2 + 1-dimensional equilibrium interface fluctuation models with periodic nonlinearities. The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed in Lacoin et al.