Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ for $$\kappa '$$ κ ′ in (4, 8) that is drawn on an independent $$\gamma $$ γ -LQG surface for $$\gamma ^2=16/\kappa '$$ γ 2 = 16 / κ ′ . The results are similar in flavor to the ones from our companion paper dealing with $$\hbox {CLE}_{\kappa }$$ CLE κ for $$\kappa $$ κ in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ independently into two colors with respective probabilities p and $$1-p$$ 1 - p . This description was complete up to one missing parameter $$\rho $$ ρ . The results of the present paper about CLE on LQG allow us to determine its value in terms of p and $$\kappa '$$ κ ′ . It shows in particular that $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ and $$\hbox {CLE}_{16/\kappa '}$$ CLE 16 / κ ′ are related via a continuum analog of the Edwards-Sokal coupling between $$\hbox {FK}_q$$ FK q percolation and the q-state Potts model (which makes sense even for non-integer q between 1 and 4) if and only if $$q=4\cos ^2(4\pi / \kappa ')$$ q = 4 cos 2 ( 4 π / κ ′ ) . This provides further evidence for the long-standing belief that $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ and $$\hbox {CLE}_{16/\kappa '}$$ CLE 16 / κ ′ represent the scaling limits of $$\hbox {FK}_q$$ FK q percolation and the q-Potts model when q and $$\kappa '$$ κ ′ are related in this way. Another consequence of the formula for $$\rho (p,\kappa ')$$ ρ ( p , κ ′ ) is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

The existence of a giant cluster for percolation on large Crump–Mode–Jagers trees

In this paper we consider random trees associated with the genealogy of Crump–Mode–Jagers processes and perform Bernoulli bond-percolation whose parameter depends on the size of the tree. Our purpose is to show the existence of a giant percolation cluster for appropriate regimes as the size grows. We stress that the family trees of Crump–Mode–Jagers processes include random recursive trees, preferential attachment trees, binary search trees for which this question has been answered by Bertoin [7], as well as (more general) m-ary search trees, fragmentation trees, and median-of-( $2\ell+1$ ) binary search trees, to name a few, where to our knowledge percolation has not yet been studied.

An overview of tools, software, and methods for natural product fragment and mass spectral analysis

One major challenge in natural product (NP) discovery is the determination of the chemical structure of unknown metabolites using automated software tools from either GC–mass spectrometry (MS) or liquid chromatography–MS/MS data only. This chapter reviews the existing spectral libraries and predictive computational tools used in MS-based untargeted metabolomics, which is currently a hot topic in NP structure elucidation. We begin by focusing on spectral databases and the general workflow of MS annotation. We then describe software and tools used in MS, particularly those used to predict fragmentation patterns, mass spectral classifiers, and tools for fragmentation trees analysis. We then round up the chapter by looking at more advanced approaches implemented in tools for competitive fragmentation modeling and quantum chemical approaches.

Generalized Markov branching trees

Motivated by the gene tree/species tree problem from statistical phylogenetics, we extend the class of Markov branching trees to a parametric family of distributions on fragmentation trees that satisfies a generalized Markov branching property. The main theorems establish important statistical properties of this model, specifically necessary and sufficient conditions under which a family of trees can be constructed consistently as sample size grows. We also consider the question of attaching random edge lengths to these trees.