Geodesic stars in random geometry

Supercritical causal maps : geodesics and simple random walk

We study the random planar maps obtained from supercritical Galton--Watson trees by adding the horizontal connections between successive vertices at each level. These are the hyperbolic analog of the maps studied by Curien, Hutchcroft and Nachmias in arXiv:1710.03137, and a natural model of random hyperbolic geometry. We first establish metric hyperbolicity properties of these maps: we show that they admit bi-infinite geodesics and satisfy a weak version of Gromov-hyperbolicity. We also study the simple random walk on these maps: we identify their Poisson boundary and, in the case where the un…

有向景观中的对偶性及其在分形几何中的应用

Duality in the directed landscape and its applications to fractal geometry

本文利用有向景观中测地线树与对偶树之间的对偶性，将测地线树非典型点集的分形问题转化为对偶树问题，证明固定方向上允许两条半无限测地线的点集几乎必然具有Hausdorff维数4/3，且允许三条的点集几乎必然可数。

Geodesics and almost geodesic cycles in random regular graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u,v) is at least d_C(u,v)-e(n). Let f(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n)= \log_{d-1} \log_{d-1} n +f(n) and |C|=2\log_{d-1}n+O(f(n)). Along the way, we obtain results on near-geodesic paths. We also give the limiting distri…

Random coalescing geodesics in first-passage percolation

We continue the study of infinite geodesics in planar first-passage percolation, pioneered by Newman in the mid 1990s. Building on more recent work of Hoffman, and Damron and Hanson, we develop an ergodic theory for infinite geodesics via the study of what we shall call `random coalescing geodesics'. Random coalescing geodesics have a range of nice asymptotic properties, such as asymptotic directions and linear Busemann functions. We show that random coalescing geodesics are (in some sense) dense in the space of geodesics. This allows us to extrapolate properties from random coalescing geodesi…

Slowdown for the geodesic-biased random walk

Given a connected graph $G$ with some subset of its vertices excited and a fixed target vertex, in the geodesic-biased random walk on $G$, a random walker moves as follows: from an unexcited vertex, she moves to a uniformly random neighbour, whereas from an excited vertex, she takes one step along some fixed shortest path towards the target vertex. We show, perhaps counterintuitively, that the geodesic-bias can slow the random walker down exponentially: there exist connected, bounded-degree $n$-vertex graphs with excitations where the expected hitting time of a fixed target is at least $\exp (…

On the geometry of geodesics in discrete optimal transport

We consider the space of probability measures on a discrete set $X$, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset $Y \subseteq X$, it is natural to ask whether they can be connected by a constant speed geodesic with support in $Y$ at all times. Our main result answers this question affirmatively, under a suitable geometric condition on $Y$ introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi equations, which is of independent interest.

Fold maps associated to geodesic random walks on non-positively curved manifolds

We study a family of mappings from the powers of the unit tangent sphere at a point to a complete Riemannian manifold with non-positive sectional curvature, whose behavior is related to the spherical mean operator and the geodesic random walks on the manifold. We show that for odd powers of the unit tangent sphere the mappings are fold maps. Some consequences on the regularity of the transition density of geodesic random walks, and on the eigenfunctions of the spherical mean operator are discussed and related to previous work.

Large deviations for geodesic random walks

We provide a direct proof of Cramér's theorem for geodesic random walks in a complete Riemannian manifold $(M,g)$. We show how to exploit the vector space structure of the tangent spaces to study large deviation properties of geodesic random walks in $M$. Furthermore, we reveal the geometric obstructions one runs into and overcome these by providing Taylor expansions of the inverse Riemannian exponential map, while also comparing the differential of the exponential map to parallel transport. Finally, we obtain the analogue of Cramér's theorem for geodesic random walks by showing that the curva…

平面Poisson道路随机度量中的测地线

Geodesics in planar Poisson roads random metric

该研究针对平面自相似Poisson道路随机度量，证明了测地线途中不使用任意低速道路、驶离道路非测地等结论，完整刻画了道路点附近的测地线局部结构，验证了Kendall的相关猜想。

Lengths of closed geodesics on random surfaces of large genus

We prove Poisson approximation results for the bottom part of the length spectrum of a random closed hyperbolic surface of large genus. Here, a random hyperbolic surface is a surface picked at random using the Weil-Petersson volume form on the corresponding moduli space. As an application of our result, we compute the large genus limit of the expected systole.

Geodesic random walks, diffusion processes and Brownian motion on Finsler manifolds

We show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

群上的随机游走与超线性发散测地线

Random walks on groups and superlinear divergent geodesics

本文研究含超线性发散测地线的群上的随机游走，利用拟等距不变量执行 Gouëzel 枢轴技术，建立中心极限定理。

Sampling by Intersections with Random Geodesics

In this paper we compare the different phenomena that occur when intersecting geometric objects with random geodesics on the unit sphere and inside convex bodies. On the high dimensional sphere we see that with probability bounded away from zero, the observed length will deviate from the actual measure by at most a fixed error for any subset, while in convex bodies we can always choose a subset for which the behavior would be close to a zero-one law, as the dimension grows. The result for the sphere is based on an analysis of the Radon transform. Using similar tools we analyze the variance of …

Self-Intersections of Random Geodesics on Negatively Curved Surfaces

We study the fluctuations of self-intersection counts of random geodesic segments of length $t$ on a compact, negatively curved surface in the limit of large $t$. If the initial direction vector of the geodesic is chosen according to the \emph{Liouville measure}, then it is not difficult to show that the number $N (t)$ of self-intersections by time $t$ grows like $κt^{2}$, where $κ=κ_{M}$ is a positive constant depending on the surface $M$. We show that (for a smooth modification of $N (t)$) the fluctuations are of size $t$, and the limit distribution is a weak limit of Gaussian quadratic form…

大亏格随机曲面上素测地线的中心极限定理

A central limit theorem for prime geodesics on random surfaces of large genus

该研究证明了大亏格Weil–Petersson随机闭双曲曲面上，范数在特定区间内的素测地线的归一化加权计数满足渐近高斯分布，拓展了素测地线计数的相关结论。

Geodesics of Random Riemannian Metrics

We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolutely continuous with respect to the law of the random metric, and we provide an explicit form for its Radon-Nikodym derivative. We use this result to prove a "local Markov property" along an unbounded geodesic, demonstrating that it eventually encounters any type of geometric phen…

超几何群上的随机测地流

Randomized Geodesic Flow on Hyperbolic Groups

本文提出随机测地流的概念，通过随机游走轨迹构造测度Θ，并证明其在所有阶数上指数混合及功能CLT。

Confluence of geodesic paths and separating loops in large planar quadrangulations

We consider planar quadrangulations with three marked vertices and discuss the geometry of triangles made of three geodesic paths joining them. We also study the geometry of minimal separating loops, i.e. paths of minimal length among all closed paths passing by one of the three vertices and separating the two others in the quadrangulation. We concentrate on the universal scaling limit of large quadrangulations, also known as the Brownian map, where pairs of geodesic paths or minimal separating loops have common parts of non-zero macroscopic length. This is the phenomenon of confluence, which …

球面上测地线随机游走的Wasserstein收缩率的维度无关上界

A Dimension-Independent Bound on the Wasserstein Contraction Rate of a Geodesic Random Walk on the Sphere

本文证明了欧几里得$d$-球面上测地线随机游走的Wasserstein收缩率具有与维度$d$无关的上界，为geodesic slice sampling和Gibbsian polar slice sampling等MCMC方法实现维度无关性能提供了理论依据。

Geodesics of Random Riemannian Metrics: Supplementary Material

This is supplementary material for the main Geodesics article by the authors. In Appendix A, we present some general results on the construction of Gaussian random fields. In Appendix B, we restate our Shape Theorem, specialized to the setting of this article. In Appendix C, we state some straightforward consequences on the geometry of geodesics for a random metric. In Appendix D, we provide a rapid introduction to Riemannian geometry for the unfamiliar reader. In Appendix E, we present some analytic estimates which we use in the article. In Appendix F, we present the construction of the condi…

Estimation of geodesic tortuosity and constrictivity in stationary random closed sets

We investigate the problem of estimating geodesic tortuosity and constrictivity as two structural characteristics of stationary random closed sets. They are of central importance for the analysis of effective transport properties in porous or composite materials. Loosely speaking, geodesic tortuosity measures the windedness of paths whereas the notion of constrictivity captures the appearance of bottlenecks resulting from narrow passages within a given materials phase. We first provide mathematically precise definitions of these quantities and introduce appropriate estimators. Then, we show st…

Approximating geodesics via random points

Given a `cost' functional $F$ on paths $γ$ in a domain $D\subset\mathbb{R}^d$, in the form $F(γ) = \int_0^1 f(γ(t),\dotγ(t))dt$, it is of interest to approximate its minimum cost and geodesic paths. Let $X_1,\ldots, X_n$ be points drawn independently from $D$ according to a distribution with a density. Form a random geometric graph on the points where $X_i$ and $X_j$ are connected when $0<|X_i - X_j|<ε$, and the length scale $ε=ε_n$ vanishes at a suitable rate. For a general class of functionals $F$, associated to Finsler and other distances on $D$, using a probabilistic form of Gamma converge…

随机双曲穿孔球面上的短测地线与小特征值

Short geodesics and small eigenvalues on random hyperbolic punctured spheres

本文研究了 $n\to\infty$ 时 Weil-Petersson 随机双曲穿孔球面上的短测地线与小特征值，证明闭测地线长度经缩放后收敛于 Poisson 点过程，并得出具有至少 $k=o(n)$ 个任意小特征值的概率趋于 $1$。