Related papers: Probability theory for random groups arising in nu…
The goal of this paper is to prove theorems that elucidate the Cohen-Lenstra-Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler…
The moment problem in probability theory asks for criteria for when there exists a unique measure with a given tuple of moments. We study a variant of this problem for random objects in a category, where a moment is given by the average…
This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result…
The Cohen-Lenstra heuristic is a universal principle that assigns to each group a probability that tells how often this group should occur "in nature". The most important, but not the only, applications are sequences of class groups, which…
Let $L/K$ be a quadratic extension of global fields. We study Cohen-Lenstra heuristics for the $\ell$-part of the relative class group $G_{L/K} := \textrm{Cl}(L/K)$ when $K$ contains $\ell^n$th roots of unity. While the moments of a…
The Cohen-Lenstra-Martinet Heuristics gives a prediction of the distribution of $\operatorname{Cl}_K[p^\infty]$ whne $K$ runs over $\Gamma$-fields and $p\nmid|\Gamma|$. In this paper, we prove several results on the distribution of ideal…
One of the most elusive challenges within the area of topological data analysis is understanding the distribution of persistence diagrams. Despite much effort, this is still largely an open problem. In this paper, we present a series of…
Let $\mathcal{M}_n(E)$ denote the set of vectors of the first $n$ moments of probability measures on $E\subset\mathbb{R}$ with existing moments. The investigation of such moment spaces in high dimension has found considerable interest in…
In number theory, great efforts have been undertaken to study the Cohen-Lenstra probability measure on the set of all finite abelian $p$-groups. On the other hand, group theorists have studied a probability measure on the set of all…
The main aim of the present paper is to disprove the Cohen--Lenstra--Martinet heuristics in two different ways and to offer possible corrections. We also recast the heuristics in terms of Arakelov class groups, giving an explanation for the…
Some tools and ideas are interchanged between random matrix theory and multivariate statistics. In the context of the random matrix theory, classes of spherical and generalised Wishart random matrix ensemble, containing as particular cases…
A random set is a generalisation of a random variable, i.e. a set-valued random variable. The random set theory allows a unification of other uncertainty descriptions such as interval variable, mass belief function in Dempster-Shafer theory…
Work on generalizations of the Cohen-Lenstra and Cohen-Martinet heuristics has drawn attention to probability measures on the space of isomorphism classes of profinite groups. As is common in probability theory, it would be desirable to…
The Cohen-Lenstra-Martinet heuristics predict the frequency with which a fixed finite abelian group appears as an ideal class group of an extension of number fields, for certain sets of extensions of a base field. Recently, Malle found…
Cohen, Lenstra, and Martinet have given conjectures for the distribution of class groups of extensions of number fields, but Achter and Malle have given theoretical and numerical evidence that these conjectures are wrong regarding the Sylow…
The Cohen-Lenstra heuristic predicts the distribution of ideal class groups over number fields. Random matrix models provide a natural framework for explaining this heuristic, and recent results demonstrate the effectiveness of these tools.…
We propose a modification to the Cohen--Lenstra prediction for the distribution of class groups of number fields, which should also apply when the base field contains non-trivial roots of unity. The underlying heuristic derives from the…
In probability theory, there is a tendency to treat one random variable with a given distribution as being just as good as any other. By and large this is fine because probability is (mostly) concerned with distributional properties of…
Motivated by recent results in random matrix theory we will study the distributions arising from products of complex Gaussian random matrices and truncations of Haar distributed unitary matrices. We introduce an appropriately general class…
In this article the idea of random variables over the set theoretic universe is investigated. We explore what it can mean for a random set to have a specific probability of belonging to an antecedently given class of sets.