Related papers: On limit theorems for persistent Betti numbers fro…

Multidimensional persistence modules do not admit a concise representation analogous to that provided by persistence diagrams for real-valued functions. However, there is no obstruction for multidimensional persistent Betti numbers to admit…

Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane $\mathbb{R}^2$ that can…

Persistent homology computes topological invariants from point cloud data. Recent work has focused on developing statistical methods for data analysis in this framework. We show that, in certain models, parametric inference can be performed…

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise).…

We develop a general framework for the probabilistic analysis of random finite point clouds in the context of topological data analysis. We extend the notion of a barcode of a finite point cloud to compact metric spaces. Such a barcode…

In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this…

Let $\mathbb{F}$ be a field, let $P \subseteq \mathbb{F}^d$ be a finite set of points, and let $\alpha,\beta \in \mathbb{F} \setminus \{0\}$. We study the quantity \[|\Pi_{\alpha, \beta}| = \{(p,q,r) \in P \times P \times P \mid p \cdot q =…

Let $Y_i,i\geq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $\mathcal {M}\subset \mathbb{R}^d$ and consider sums $\sum_{i=1}^n\xi(n^{1/m}Y_i,\{n^{1/m}Y_j\}_{j=1}^n)$, where $\xi$ is a real valued function…

Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those…

For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random…

Let $P$ be a simple,stationary point process having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $P_n:= P \cap W_n$ be its…

Phase transition in the two-dimensional $q$-state Potts model with random ferromagnetic couplings in the large-q limit is conjectured to be described by the isotropic version of the infinite randomness fixed point of the random…

For $\Delta \ge 5$ and $q$ large as a function of $\Delta$, we give a detailed picture of the phase transition of the random cluster model on random $\Delta$-regular graphs. In particular, we determine the limiting distribution of the…

We start with a simple introduction to topological data analysis where the most popular tool is called a persistent diagram. Briefly, a persistent diagram is a multiset of points in the plane describing the persistence of topological…

This paper studies beta ensembles on the real line in a high temperature regime, that is, the regime where $\beta N \to const \in (0, \infty)$, with $N$ the system size and $\beta$ the inverse temperature. In this regime, the convergence to…

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ``hubs''. We study the higher-order connectivity of such a network by investigating the topological…

Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters -- the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical…

Drees and Rootz\'en [2010] have proven central limit theorems (CLT) for empirical processes of extreme values cluster functionals built from $\beta$-mixing processes. The problem with this family of $\beta$-mixing processes is that it is…

We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together…

Let X be a k-dimensional simplicial complex such that the (k-j-2)-dimensional homology of the links of all j-dimensional simplices in X vanishes. An upper bound is given on the (k-1)-th Betti number of X. Examples based on sum complexes…