Decomposition of Levy trees along their diameter

We study the diameter of L{\'e}vy trees that are random compact metric spaces obtained as the scaling limits of Galton-Watson trees. L{\'e}vy trees have been introduced by Le Gall and Le Jan (1998) and they generalise Aldous' Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of L{\'e}vy trees and we prove that it is realized by a unique pair of points. We prove that the law of L{\'e}vy trees conditioned to have a fixed diameter r $\in$ (0, $\infty$) is obtained by glueing at their respective roots two independent size-biased L{\'e}vy trees conditioned to have height r/2 and then by uniformly re-rooting the resulting tree; we also describe by a Poisson point measure the law of the subtrees that are grafted on the diameter. As an application of this decomposition of L{\'e}vy trees according to their diameter, we characterize the joint law of the height and the diameter of stable L{\'e}vy trees conditioned by their total mass; we also provide asymptotic expansions of the law of the height and of the diameter of such normalised stable trees, which generalises the identity due to Szekeres (1983) in the Brownian case

Height and diameter of Brownian tree

By computations on generating functions, Szekeres proved in 1983 that the law of the diameter of a uniformly distributed rooted labelled tree with n vertices, rescaled by a factor n^{−1/2}, converges to a distribution whose density is explicit. Aldous observed in 1991 that this limiting distribution is the law of the diameter of the Brownian tree. In our article, we provide a computation of this law which is directly based on the normalized Brownian excursion. Moreover, we provide an explicit formula for the joint law of the height and diameter of the Brownian tree, which is a new result

Decomposition of Lévy trees along their diameter

We study the diameter of Lévy trees that are random compact metric spaces obtained as the scaling limits of Galton–Watson trees. Lévy trees have been introduced by Le Gall & Le Jan (Ann. Probab. 26 (1998) 213–252) and they generalise Aldous’ Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of Lévy trees and we prove that it is realized by a unique pair of points. We prove that the law of Lévy trees conditioned to have a fixed diameter r ∈ (0, ∞) is obtained by glueing at their respective roots two independent size-biased Lévy trees conditioned to have height r/2 and then by uniformly re-rooting the resulting tree; we also describe by a Poisson point measure the law of the subtrees that are grafted on the diameter. As an application of this decomposition of Lévy trees according to their diameter, we characterize the joint law of the height and the diameter of stable Lévy trees conditioned by their total mass; we also provide asymptotic expansions of the law of the height and of the diameter of such normalised stable trees, which generalises the identity due to Szekeres (In Combinatorial Mathematics, X (Adelaide, 1982) (1983) 392–397 Springer) in the Brownian case

Cutting down p-trees and inhomogeneous continuum random trees

We study a fragmentation of the p-trees of Camarri and Pitman. We give exact correspondences between the p-trees and trees which encode the fragmentation. We then use these results to study the fragmentation of the inhomogeneous continuum random trees (scaling limits of p-trees) and give distributional correspondences between the initial tree and the tree encoding the fragmentation. The theorems for the inhomogeneous continuum random tree extend previous results by Bertoin and Miermont about the cut tree of the Brownian continuum random tree

Reversing the cut tree of the Brownian continuum random tree

Consider the Aldous–Pitman fragmentation process of a Brownian continuum random tree T^{br}. The associated cut tree cut(T^{br}), introduced by Bertoin and Miermont, is defined in a measurable way from the fragmentation process, describing the genealogy of the fragmentation, and is itself distributed as a Brownian CRT. In this work, we introduce a shuffle transform, which can be considered as the reverse of the map taking T br to cut(T^{br})

A new combinatorial representation of the additive coalescent

The standard additive coalescent starting with n particles is a Markov process which owns several combinatorial representations, one by Pitman as a process of coalescent forests, and one by Chassaing and Louchard as the block sizes in a parking scheme. In the coalescent forest representation, edges are added successively between a random node and a random root. In this paper, we investigate an alternative construction by, instead, adding edges between roots. This construction induces exactly the same process in terms of cluster sizes, meanwhile, it allows us to make numerous new connections with other combinatorial and probabilistic models: size biased percolation, parking scheme in a tree, increasing trees, random cuts of trees. The variety of the combinatorial objects involved justifies our interest in this construction

Extended Depth-range Dual-wavelength Interferometry Based on Iterative Two-step Temporal Phase-unwrapping

Phase retrieval is one of the most challenging processes in many interferometry techniques. To promote the phase retrieval, Xu et. al [X. Xu, Y. Wang, Y. Xu, W. Jin. 2016] proposed a method based on dual-wavelength interferometry. However, the phase-difference brings large noise due to its low sensitivity and signal-to-noise ratio (SNR). Beside, special phase shifts are required in Xu's method. In the light of these problems, an extended depth-range dual-wavelength phase-shifting interferometry is proposed. Firstly, the least squares algorithm is utilized to retrieve the single-wavelength phase from a sequence of N-frame simultaneous phase-shifting dual-wavelength interferograms (SPSDWI) with random phase shifts. Then the phase-difference and phase-sum are calculated from the wrapped phases of single wavelength, and the iterative two-step temporal phase-unwrapping is introduced to unwrap the phase-sum, which can extend the depth-range and improve the sensitivity. Finally, the height of objects is achieved. Simulated experiments are conducted to demonstrate the superb precision and overall performance of the proposed method.Comment: 21 pages, 19 figure