From random lines to metric spaces (Q516126)

First, the author defines a scale-invariant random spatial networks (SIRSN) (see [\textit{D. Aldous}, Electron. J. Probab. 19, Paper No. 15, 41 p. (2014; Zbl 1305.90104)]). Consider a \(d\)-dimensional random mechanism, which provides random routes connecting any two points \(x_1\), \(x_2\in\mathbb R^d\). We say that this is a SIRSN if the following properties hold:{\parindent = 0.7 cm \begin{itemize}\item[(1)] Between any specified two points \(x_1\), \(x_2\in \mathbb R^d\), almost surely the random mechanism provides just one connecting random route \(R(x_1, x_2)=R(x_2, x_1)\), which is a finite-length path connecting \(x_1\) to \(x_2\). \item[(2)] For a finite set of points \(x_1, \ldots, x_k\in\mathbb R^d\), consider the random network \(N(x_1, \ldots , x_k)\) formed by random routes provided by the structure to connect all \(x_i\) and \(x_j\). Then \(N(x_1, \ldots , x_k)\) is statistically invariant under translation, rotation, and rescaling: if \(\sigma\) is a Euclidean similarity of \(\mathbb R^d\) then the networks \(\sigma N(x_1, \ldots , x_k)\) and \(N(\sigma x_1, \ldots , \sigma x_k)\) have the same distribution. \item[(3)] Let \(D_1\) be the length of the route between two points separated by unit Euclidean distance. Then \(E[D_1]<\infty\). \item[(4)] Suppose that \(\Xi_\lambda\) is a Poisson point process in \(\mathbb R^d\), of intensity \(\lambda>0\) and independent of the random mechanism in question. Then \(N(\Xi_\lambda)\), the union of all the networks \(N(x_1, \ldots , x_k)\) for \(x_1, \ldots , x_k\in \Xi_\lambda\), is a locally finite fiber process in \(\mathbb R^d\). That is to say, for any compact set \(K\) the total length of \(N(\Xi_\lambda)\cap K\) is almost surely finite. \item[(5)] The length intensity \(l\) of \(N(\Xi_1)\) (the mean length per unit area) is finite. \item[(6)] Suppose the Poisson point process \(\{\Xi_\lambda:\, \lambda>0\}\) are coupled so that \(\Xi_{\lambda_1} \subseteq \Xi_{\lambda_2}\) if \(\lambda_1<\lambda_2\). The fiber process \(\bigcup_{\lambda>0} \bigcup_{x_1, x_2\in \Xi_{\lambda}} (R(x_1, x_2)\setminus (\text{ball}(x_1, 1)\cup (\text{ball}(x_2, 1)))\) has length intensity bounded above by a finite constant \(p(1)\). \end{itemize}} If only properties (1)--(5) are satisfied, then the random mechanism is called a weak SIRSN. If only properties (1)--(4) are satisfied, then the random mechanism is called a pre-SIRSN. The purpose of this paper is to explore the Poisson line process model: it is at least a pre-SIRSN if \(d=2\) and if \(d>2\) the construction provides a random metric space in \(\mathbb R^d\). The chief difficulty in analyzing any of these random mechanism lies in the fact that it is hard to work with explicit minimum-time paths, whose explicit construction would involve solving a nonlocal minimization problem to determine geodesics. The structure of the paper is as follows. The rest of the introduction is concerned with basic notions of stochastic geometry and with the definition of the underlying improper Poisson line process \(\Pi\) marked with speeds. This improper Poisson line process \(\Pi\) is defined by an intensity measure \((\gamma-1) \nu^{-\gamma} d\nu \mu_d(dl)\) (for speed \(\nu>0\), parameter \(\gamma>1\), and invariant measure \(\mu_d\) on line space) and supplies a measurable orientation field marked by speeds. Section 2 then explores the way in which the measurable orientation field can be integrated to provide Lipschitz paths based on the marked line process, namely \(\Pi\)-paths. Sobolev space and comparison arguments can then be used to establish a priori bounds on Lipschitz constants for finite-time \(\Pi\)-paths, hence closure, weak closure and finally weak compactness of finite-time \(\Pi\)-paths. All these results require \(\gamma\geq d\). Note that dimension \(d>1\) is required if the line-process theory is to be nontrivial. Section 3 shows that, given \(\gamma>d\) and fixed points \(x_1, x_2\in\mathbb R^d\), it is almost surely possible to connect \(x_1\) to \(x_2\) in finite time with \(\Pi\)-paths, and indeed with probability 1 it is possible to connect all pairs of points in this way. This implies the existence of minimum-time \(\Pi\)-paths, namely \(\Pi\)-geodesics. In dimension \(d>2\), this is a rather unexpected result, since almost surely none of the lines of \(\Pi\) will then intersect. Nevertheless, \(\Pi\) then furnishes \(\mathbb R^d\) with the structure of a random geodesic might look like. In particular, these results imply measurability of the random time taken to pass from one point to another using a \(\Pi\)-geodesic. The reminder of the paper is restricted to the planar case of \(d=2\), since the arguments now make essential use of point-line duality. Consider the extent to which networks formed by \(\Pi\)-geodesics fulfill the requirements of the definition. The statistical invariance property 2 follows immediately from similar invariance of the underlying intensity measure of the improper Poisson line process (whether planar or not). Property 1 requires almost sure uniqueness of network routes. Section 4 establishes this for \(\gamma>d=2\), using a careful analysis of the nature of planar \(\Pi\)-geodesics which falls just short of establishing that planar \(\Pi\)-geodesics can be made up of consecutive sequences of line segments. While \(\Pi\)-geodesics between pairs of points are minimum-time paths, the fact that they have finite mean length is not immediately apparent; this is established in Section 5, first for restricted planar \(\Pi\)-geodesics, then for general planar \(\Pi\)-geodesics. Thus the finite-mean-length Property 3 of the definition is verified for \(d=2\). Finally the pre-SIRSN Property 6 is established for the planar case in Section 6, here also is established the uniqueness of planar \(\Pi\)-geodesics reaching out of infinity and, for any specified point \(x\in\mathbb R^d\), the fact that all \(\Pi\)-geodesics emanating from \(x\) must coincide for initial periods. These results are established using an essentially soft argument concerning the existence of certain structures in \(\Pi\). The concluding Section 7 notes that more quantitative arguments would be required to decide whether the weak SIRSN or full SIRSN properties hold. Section 7 also notes some other interesting open questions.