Abstract
The dead leaves model (DLM) provides a random tessellation of $d$-space, representing the visible portions of fallen leaves on the ground when $d=2$. For $d=1$, we establish formulae for the intensity, two-point correlations, and asymptotic covariances for the point process of cell boundaries, along with a functional CLT. For $d=2$ we establish analogous results for the random surface measure of cell boundaries, and also determine the intensity of cells in a more general setting than in earlier work of Cowan and Tsang. We introduce a general notion of dead leaves random measures and give formulae for means, asymptotic variances and functional CLTs for these measures; this has applications to various other quantities associated with the DLM.
Highlights
The dead leaves model in d-dimensional space (d ∈ N), due originally to Matheron [21], is defined as follows [4, 34]
As well as the new results already mentioned, we provide some extensions to known first-order results, giving the intensity of cell boundaries in d = 1, and the intensity of cells in d = 2
We review some concepts from the theory of point processes and random measures that we shall be using
Summary
The dead leaves model (or DLM for short) in d-dimensional space (d ∈ N), due originally to Matheron [21], is defined as follows [4, 34]. Note that in the present paper, all cells of our DLM tessellation are connected; the tessellation where cells are taken to be the visible portions of leaves (rather than their connected components) is of interest, and is considered in some of the works just mentioned. We develop (in Section 4) an extension of the DLM which we call the dead leaves random measure (DLRM). Since we prefer to work with positive rather than negative times, in this paper we often consider a time-reversed version of the DLM where, for each site x ∈ Rd, the first leaf to arrive at x after time 0 is taken to be visible at x. Glass plate which can be observed from below, starting from time 0 This gives a tessellation with the same distribution as the original DLM.
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