Anomalous diffusion of random walk on random planar maps

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.

Let γ ∈ ( 0 , 2 ) \gamma \in (0,2) , let h h be the planar Gaussian free field, and let D h D_h be the associated γ \gamma -Liouville quantum gravity (LQG) metric. We prove that for any random Borel set X ⊂ C X \subset \mathbb {C} which is independent from h h , the Hausdorff dimensions of X X with respect to the Euclidean metric and with respect to the γ \gamma -LQG metric D h D_h are a.s. related by the (geometric) KPZ formula. As a corollary, we deduce that the Hausdorff dimension of the continuum γ \gamma -LQG metric is equal to the exponent d γ > 2 d_\gamma > 2 studied by Ding and Gwynne (2018), which describes distances in discrete approximations of γ \gamma -LQG such as random planar maps. We also derive “worst-case” bounds relating the Euclidean and γ \gamma -LQG dimensions of X X when X X and h h are not necessarily independent, which answers a question posed by Aru (2015). Using these bounds, we obtain an upper bound for the Euclidean Hausdorff dimension of a γ \gamma -LQG geodesic which equals 1.312 … 1.312\dots when γ = 8 / 3 \gamma = \sqrt {8/3} ; and an upper bound of 1.9428 … 1.9428\dots for the Euclidean Hausdorff dimension of a connected component of the boundary of a 8 / 3 \sqrt {8/3} -LQG metric ball. We use the axiomatic definition of the γ \gamma -LQG metric, so the paper can be understood by readers with minimal background knowledge beyond a basic level of familiarity with the Gaussian free field.

Let M be the infinite spanning-tree-weighted random planar map, which is the local limit of finite random planar maps sampled with probability proportional to the number of spanning trees they admit. We show that a.s. the M-graph-distance diameter of the external diffusion-limited aggregation (DLA) cluster on M run for m steps is of order m2/d+om(1), where d is the metric ball volume growth exponent for M (which was shown to exist by Ding and Gwynne (Comm. Math. Phys. 374 (2020) 1877–1934). By known bounds for d, one has 0.55051⋯≤2/d≤0.563315…. Along the way, we also prove that loop-erased random walk (LERW) on M typically travels graph distance m2/d+om(1) in m units of time and that the graph-distance diameter of a finite spanning-tree-weighted random planar map with n edges, with or without boundary, is of order n1/d+on(1) except on an event with probability decaying faster than any negative power of n. Our proofs are based on a special relationship between DLA and LERW on spanning-tree-weighted random planar maps as well as estimates for distances in such maps which come from the theory of Liouville quantum gravity.

We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embeddings) of certain random planar maps converge to γ-Liouville quantum gravity (γ-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of correlated continuum random trees, and γ ranges from 0 to 2 as one varies the correlation parameter. We also show that the associated space-filling path on the embedded map converges to space-filling SLEκ for κ=16/γ2 (in the annealed sense) and that simple random walk on the embedded map converges to Brownian motion (in the quenched sense). This work constitutes the first proof that a discrete conformal embedding of a random planar map converges to LQG. Many more such statements have been conjectured. Since the mated-CRT map can be viewed as a coarse-grained approximation to other random planar maps (the UIPT, tree-weighted maps, bipolar-oriented maps, etc.), our results indicate a potential approach for proving that embeddings of these maps converge to LQG as well. To prove the main result, we establish several (independently interesting) theorems about LQG surfaces decorated by space-filling SLE. There is a natural way to use the SLE curve to divide the plane into “cells” corresponding to vertices of the mated-CRT map. We study the law of the shape of the origin-containing cell, in particular proving moments for the ratio of its squared diameter to its area. We also give bounds on the degree of the origin-containing cell and establish a form of ergodicity for the entire configuration. Ultimately, we use these properties to show (with the help of a general theorem proved in a separate paper) that random walk on these cells converges to a time change of Brownian motion, which in turn leads to the Tutte embedding result.

For γ∈(0,2), the quantum disk and γ-quantum wedge are two of the most natural types of Liouville quantum gravity (LQG) surfaces with boundary. These surfaces arise as scaling limits of finite and infinite random planar maps with boundary, respectively. We show that the left/right quantum boundary length process of a space-filling SLE16/γ2 curve on a quantum disk or on a γ-quantum wedge is a certain explicit conditioned two-dimensional Brownian motion with correlation −cos(πγ2/4). This extends the mating of trees theorem of Duplantier, Miller, and Sheffield (2014) to the case of quantum surfaces with boundary (the disk case for γ∈(2,2) was previously treated by Duplantier, Miller, Sheffield using different methods). As an application, we give an explicit formula for the conditional law of the LQG area of a quantum disk given its boundary length by computing the law of the corresponding functional of the correlated Brownian motion.

In this paper, we construct Liouville Quantum Field Theory (LQFT) on the toroidal topology in the spirit of the 1981 seminal work by Polyakov [Phys. Lett. B 103, 207 (1981)]. Our approach follows the construction carried out by the authors together with Kupiainen in the case of the Riemann sphere [“Liouville quantum gravity on the Riemann sphere,” e-print arXiv:1410.7318]. The difference is here that the moduli space for complex tori is non-trivial. Modular properties of LQFT are thus investigated. This allows us to integrate the LQFT on complex tori over the moduli space, to compute the law of the random Liouville modulus, therefore recovering (and extending) formulae obtained by physicists, and make conjectures about the relationship with random planar maps of genus one, eventually weighted by a conformal field theory and conformally embedded onto the torus.

The Distribution of the Maximum Vertex Degree in Random Planar Maps

We study the survival probability and the growth rate for branching random walks in random environment (BRWRE). The particles perform simple symmetric random walks on the d-dimensional integer lattice, while at each time unit, they split into independent copies according to time–space i.i.d. offspring distributions. The BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity called the free energy is well studied. We discuss the survival probability (both global and local) for BRWRE and give a criterion for its positivity in terms of the free energy of the associated DPRE. We also show that the global growth rate for the number of particles in BRWRE is given by the free energy of the associated DPRE, though the local growth rate is given by the directional free energy.

Pattern occurrences in random planar maps

We study the geodesic two- and three-point functions of random weighted cubic maps, which are obtained by assigning random edge lengths to random cubic planar maps. Explicit expressions are obtained by taking limits of recently established bivariate multi-point functions of general planar maps. We give an alternative interpretation of the two-point function in terms of an Eden model exploration process on a random planar triangulation. Finally, the scaling limits of the multi-point functions are studied, showing in particular that the two- and three-point functions of the Brownian map are recovered as the number of faces is taken to infinity.

We revisit the sharp Sobolev inequalities involving boundary terms on Riemannian manifolds with boundaries proved by Li and Zhu (Geom Funct Anal 8: 59–87, 1998) and explore the role of the mean curvature.

We discuss new proofs, and new forms, of a reverse logarithmic Sobolev inequality, with respect to the standard Gaussian measure, for low-complexity functions, measured in terms of Gaussian-width. In particular, we provide a dimension-free improvement for a related result given in Eldan (Geom Funct Anal 28:1548–1596, 2018).

Let ( M , g ) be a smooth n-dimensional compact Riemannian manifold with totally geodesic boundary ∂ M , n ≥ 2 . We prove a sharp Sobolev inequality of W 1 , p ( M ) for all 1<p<n. When p = 2 and n ≥ 3 , this inequality was proved by Li-Zhu in [Sharp Sobolev inequalities involving boundary terms. Geom Funct Anal. 1998;8:59–87].

The infinite discrete stable Boltzmann maps are "heavy-tailed" generalisations of the well-known Uniform Infinite Planar Quadrangulation. Very efficient tools to study these objects are Markovian step-by-step explorations of the lattice called peeling processes. Such a process depends on an algorithm which selects at each step the next edge where the exploration continues. We prove here that, whatever this algorithm, a peeling process always reveals about the same portion of the map, thus growing roughly metric balls. Applied to well-designed algorithms, this easily enables us to compare distances in the map and in its dual, as well as to control the so-called pioneer points of the simple random walk, both on the map and on its dual.

In this paper we study typical distances in the configuration model, when the degrees have asymptotically infinite variance. We assume that the empirical degree distribution follows a power law with exponent tau in (2,3), up to value n^{{beta _n}} for some {beta _n}gg (log n)^{-gamma } and gamma in (0,1). This assumption is satisfied for power law i.i.d. degrees, and also includes truncated power-law empirical degree distributions where the (possibly exponential) truncation happens at n^{{beta _n}}. These examples are commonly observed in many real-life networks. We show that the graph distance between two uniformly chosen vertices centers around 2 log log (n^{{beta _n}}) / |log (tau -2)| + 1/({beta _n}(3-tau )), with tight fluctuations. Thus, the graph is an ultrasmall world whenever 1/{beta _n}=o(log log n). We determine the distribution of the fluctuations around this value, in particular we prove these form a sequence of tight random variables with distributions that show log log -periodicity, and as a result it is non-converging. We describe the topology and number of shortest paths: We show that the number of shortest paths is of order n^{f_n{beta _n}}, where f_n in (0,1) is a random variable that oscillates with n. We decompose shortest paths into three segments, two ‘end-segments’ starting at each of the two uniformly chosen vertices, and a middle segment. The two end-segments of any shortest path have length log log (n^{{beta _n}}) / |log (tau -2)|+tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length 1/({beta _n}(3-tau ))+tight, and it contains only vertices with degree at least of order n^{(1-f_n){beta _n}}, thus all the degrees on this segment are comparable to the maximal degree. Our theorems also apply when instead of truncating the degrees, we start with a configuration model and we remove every vertex with degree at least n^{{beta _n}}, and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees.