Xin Sun

Bio: Xin Sun is an academic researcher from Columbia University. The author has contributed to research in topics: Scaling limit & Random walk. The author has an hindex of 19, co-authored 64 publications receiving 865 citations. Previous affiliations of Xin Sun include Massachusetts Institute of Technology & University of Pennsylvania.

Fractional Gaussian fields: A survey

Abstract: We discuss a family of random fields indexed by a parameter $s\in\mathbb{R} $ which we call the fractional Gaussian fields, given by \[\mathrm{FGF}_{s}(\mathbb{R} ^{d})=(-\Delta)^{-s/2}W, \] where $W$ is a white noise on $\mathbb{R}^{d}$ and $(-\Delta)^{-s/2}$ is the fractional Laplacian. These fields can also be parameterized by their Hurst parameter $H=s-d/2$. In one dimension, examples of $\mathrm{FGF}_{s}$ processes include Brownian motion ($s=1$) and fractional Brownian motion ($1/2

Mating of trees for random planar maps and Liouville quantum gravity: a survey

Abstract: We survey the theory and applications of mating-of-trees bijections for random planar maps and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield (2014). The latter theorem gives an encoding of a Liouville quantum gravity (LQG) surface decorated by a Schramm-Loewner evolution (SLE) curve in terms of a pair of correlated linear Brownian motions. We assume minimal familiarity with the theory of SLE and LQG. Mating-of-trees theory enables one to reduce problems about SLE and LQG to problems about Brownian motion and leads to deep rigorous connections between random planar maps and LQG. Applications discussed in this article include scaling limit results for various functionals of decorated random planar maps, estimates for graph distances and random walk on (not necessarily uniform) random planar maps, computations of the Hausdorff dimensions of sets associated with SLE, scaling limit results for random planar maps conformally embedded in the plane, and special symmetries for $\sqrt{8/3}$-LQG which allow one to prove its equivalence with the Brownian map.

A mating-of-trees approach for graph distances in random planar maps

Abstract: We introduce a general technique for proving estimates for certain random planar maps which belong to the $$\gamma $$ -Liouville quantum gravity (LQG) universality class for $$\gamma \in (0,2)$$ . The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; $$\gamma =\sqrt{8/3}$$ ); and planar maps weighted by the number of different spanning trees ( $$\gamma =\sqrt{2}$$ ), bipolar orientations ( $$\gamma =\sqrt{4/3}$$ ), or Schnyder woods ( $$\gamma =1$$ ) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of $$\gamma $$ -LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map M to a mated-CRT map—a random planar map constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41–108, 1998) of the encoding walk for M and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in M from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when $$\gamma =\sqrt{8/3}$$ , we instead deduce estimates for the $$\sqrt{8/3}$$ -mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.

A distance exponent for Liouville quantum gravity

Abstract: Let $$\gamma \in (0,2)$$ and let h be the random distribution on $$\mathbb C$$ which describes a $$\gamma $$ -Liouville quantum gravity (LQG) cone. Also let $$\kappa = 16/\gamma ^2 >4$$ and let $$\eta $$ be a whole-plane space-filling SLE $$_\kappa $$ curve sampled independent from h and parametrized by $$\gamma $$ -quantum mass with respect to h. We study a family $$\{\mathcal G^\epsilon \}_{\epsilon >0}$$ of planar maps associated with $$(h, \eta )$$ called the LQG structure graphs (a.k.a. mated-CRT maps) which we conjecture converge in probability in the scaling limit with respect to the Gromov–Hausdorff topology to a random metric space associated with $$\gamma $$ -LQG. In particular, $$\mathcal G^\epsilon $$ is the graph whose vertex set is $$\epsilon \mathbb Z$$ , with two such vertices $$x_1,x_2\in \epsilon \mathbb Z$$ connected by an edge if and only if the corresponding curve segments $$\eta ([x_1-\epsilon , x_1])$$ and $$\eta ([x_2-\epsilon ,x_2])$$ share a non-trivial boundary arc. Due to the peanosphere description of SLE-decorated LQG due to Duplantier et al. (Liouville quantum gravity as a mating of trees, 2014), the graph $$\mathcal G^\epsilon $$ can equivalently be expressed as an explicit functional of a correlated two-dimensional Brownian motion, so can be studied without any reference to SLE or LQG. We prove non-trivial upper and lower bounds for the cardinality of a graph-distance ball of radius n in $$\mathcal G^\epsilon $$ which are consistent with the prediction of Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) for the Hausdorff dimension of LQG. Using subadditivity arguments, we also prove that there is an exponent $$\chi > 0$$ for which the expected graph distance between generic points in the subgraph of $$\mathcal G^\epsilon $$ corresponding to the segment $$\eta ([0,1])$$ is of order $$\epsilon ^{-\chi + o_\epsilon (1)}$$ , and this distance is extremely unlikely to be larger than $$\epsilon ^{-\chi + o_\epsilon (1)}$$ .

Convergence of uniform triangulations under the Cardy embedding

Abstract: We consider an embedding of planar maps into an equilateral triangle $\Delta$ which we call the Cardy embedding. The embedding is a discrete approximation of a conformal map based on percolation observables that are used in Smirnov's proof of Cardy's formula. Under the Cardy embedding, the planar map induces a metric and an area measure on $\Delta$ and a boundary measure on $\partial \Delta$. We prove that for uniformly sampled triangulations, the metric and the measures converge jointly in the scaling limit to the Brownian disk conformally embedded into $\Delta$ (i.e., to the $\sqrt{8/3}$-Liouville quantum gravity disk). As part of our proof, we prove scaling limit results for critical site percolation on the uniform triangulations, in a quenched sense. In particular, we establish the scaling limit of the percolation crossing probability for a uniformly sampled triangulation with four boundary marked points.

Conformal weldings of random surfaces: SLE and the quantum gravity zipper

Abstract: We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm–Loewner evolution (SLE), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path-decorated) random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.