度量空间中的一点若恰为 k 条互不相交测地线的端点，且这些路径中任意两条的拼接仍是一条测地线，则称为 k-枢纽。我们证明，在 Brownian 球面中，对于 k≥3 不存在 k-枢纽。
英文摘要：
A point of a metric space is called a $k$-hub if it is the endpoint of exactly $k$ disjoint geodesics, and that the concatenation of any two of these paths is still a geodesic. We prove that in the Brownian sphere, there is no $k$-hub for $k\geq 3$.
We prove the existence of semi-infinite geodesics for Brownian last-passage percolation (BLPP). Specifically, on a single event of probability one, there exist semi-infinite geodesics started from every space-time point and traveling in every asymptotic direction. Properties of these geodesics include uniqueness for a fixed initial point and direction, non-uniqueness for fixed direction but random initial points, and coalescence of all geodesics traveling in a common, fixed direction. Along the way, we prove that for fixed northeast and southwest directions, there almost surely exist no bi-inf…
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.
A point of a metric space is called a geodesic star with $m$ arms if it is the endpoint of $m$ disjoint geodesics. For every $m\in\{1,2,3,4\}$, we prove that the set of all geodesic stars with $m$ arms in the Brownian sphere has dimension $5-m$. This complements recent results of Miller and Qian, who proved that this dimension is smaller than or equal to $5-m$.
The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy$_2$ process, describes after the subtraction of a parabola the limiting law of the scaled energy of a geodesic running from the origin to a variable point on an anti-diagonal line in such problems as Poissonian last passage percolation. The ensemble of curves resulting from the Airy line ensemble after the subtraction of the same para…
We endow the $\sqrt{8/3}$-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere ${\mathbf S}^2$. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric $d_{\mathcal Q}$ on a countable dense subset of ${\mathbf S}^2$. Here we show that $d_{\mathcal Q}$ a.s. extends uniquely and continuously to a metric $\bar{d}_{\mathcal Q}$ on all of ${\mathbf S}^2$. Letting $d$ …
The directed landscape constructed in (Dauvergne-Ortmann-Virag '18) produces a directed, planar, random geometry, and is believed to be the universal scaling limit of two-dimensional first and last passage percolation models in the Kardar-Parisi-Zhang (KPZ) universality class. Geodesics in this random geometry form an important class of random continuous curves exhibiting fluctuation theory quite different from that of Brownian motion. In this vein, counterpart to Brownian local time, BLT (a self-similar measure supported on the set of zeros of Brownian motion), a local time for geodesics, GLT…
Asymptotic homology of the quotient of $PSL_2(\BR)$ by a modular group
考虑$ G:= PSL_2(\R)\equiv T^1\H^2$，一个模群$ Γ$和齐性空间$ Γ\sm G \equiv T^1(Γ\sm\H^2)$。赋予$ G $，然后是$ Γ\sm G $一个正则左不变度量，从而使其具有准双曲几何。围绕$ Γ\sm G $的柄和尖的缠绕通过$ Γ\sm G $的封闭1-形式的积分来计算。主要结果表示，在布朗和测地线的情况下，联合收敛的法律，这些积分，与应力之间的渐近独立慢，快绕组。$ Γ\sm G $的非双曲性导致了布朗渐近行为和测地线渐近行为之间的差异，这种差异不存在于黎曼曲面$Γ\sm\H^2$的水平上（通常在双曲情况下）。识别的上同调类的封闭1-形式和调和的1-形式，和等距分布的大测地球，也解决了。
In this paper we study the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) introduced in [4] by continuous time random walks on square lattices. The state space of BMVD contains a $2$-dimensional component, a $3$-dimensional component, and a "darning point" which joins these two components. Such a state space is equipped with the geodesic distance, under which BMVD is a diffusion process. In this paper, we prove that BMVD restricted on a bounded domain containing the darning point is the weak limit of continuous time reversible random walks with exponent…
We consider directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times on the vertices. A topic of great interest is the coupling structure of the weights of geodesics as the endpoints are varied spatially and temporally. A particular specialization is when one considers geodesics to points varying in the time direction starting from a given initial data. This paper considers the flat initial condition which corresponds to line-to-point last passage times. Settling a conjecture by Ferrari and Spohn (SIGMA, 2016), we show that for the passage times from the line $x+y=0$ …
The Directed Landscape, a random directed metric on the plane (where the first and the second coordinates are termed spatial and temporal respectively), was constructed in the breakthrough work of Dauvergne, Ortmann, and Virág, and has since been shown to be the scaling limit of various integrable models of Last Passage percolation, a central member of the Kardar-Parisi-Zhang universality class. It exhibits several scale invariance properties making it a natural source of rich fractal behavior. Such a study was initiated in Basu-Ganguly-Hammond, where the difference profile i.e., the differenc…
Recent work has shown that for $γ\in (0,2)$, a Liouville quantum gravity (LQG) surface can be endowed with a canonical metric. We prove several results concerning geodesics for this metric. In particular, we completely classify the possible networks of geodesics from a typical point on the surface to an arbitrary point on the surface, as well as the types of networks of geodesics joining two points which occur for a dense set of pairs of points on the surface. This latter result is the $γ$-LQG analog of the classification of geodesic networks in the Brownian map due to Angel, Kolesnik, and Mie…
There has recently been much activity within the Kardar-Parisi-Zhang universality class spurred by the construction of the canonical limiting object, the parabolic Airy sheet $\mathcal{S}:\mathbb{R}^2\to\mathbb{R}$ [arXiv:1812.00309]. The parabolic Airy sheet provides a coupling of parabolic Airy$_2$ processes -- a universal limiting geodesic weight profile in planar last passage percolation models -- and a natural goal is to understand this coupling. Geodesic geometry suggests that the difference of two parabolic Airy$_2$ processes, i.e., a difference profile, encodes important structural inf…
Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter $γ\in (0,2)$. We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (2020) gives a form…
Within the Kardar-Parisi-Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work arXiv:1812.00309 of Dauvergne, Ortmann, and Virág, this object was constructed and shown to be the limit after parabolic correction of one such model: Brownian last passage percolation. This limit object, called the directed landscape, admits geodesic paths between any two space-time points $(x,s)$ and $(y,t)$ with $s<t$. In this article, we examine fractal properties of the set of these paths. Our main results concern…
