Related papers: Large deviations for subcomplex counts and Betti n…
Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion…
We study cluster sizes in supercritical $d$-dimensional inhomogeneous percolation models with long-range edges -- such as long-range percolation -- and/or heavy-tailed degree distributions -- such as geometric inhomogeneous random graphs…
We statistically compare the relationships between frequencies of digits in continued fraction expansions of typical rational points in the unit interval and higher dimensional generalisations. This takes the form of a Large Deviation and…
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…
We consider the Erd\"{o}s--R\'{e}nyi random graph $G_{n,p}$ and we analyze the simple irreversible epidemic process on the graph, known in the literature as bootstrap percolation. We give a quantitative version of some results by Janson et…
Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they…
We study the homological properties of $\Delta_{\mathbf{r}}(n_1, \dots, n_e)$, a simplicial complex formed by sequentially gluing complete graphs along $(r_i-1)$-simplices. This construction generates precisely the chordal clique complexes,…
In this study, we delve into the discrete TC of surjective simplicial fibrations, aiming to unravel the interplay between topological complexity, discrete geometric structures, and computational efficiency. Moreover, we examine the…
We study random, finite-dimensional, ungraded chain complexes over a finite field and show that for a uniformly distributed differential a complex has the smallest possible homology with the highest probability: either zero or…
We prove a number of double-sided estimates relating discrete counterparts of several classical conformal invariants of a quadrilateral: cross-ratios, extremal lengths and random walk partition functions. The results hold true for any…
Fix a field $k$. When $\Delta$ is a simplicial complex on $n$ vertices with Stanley-Reisner ideal $I_\Delta$, we define and study an invariant called the $\textit{type defect}$ of $\Delta$. Except when $\Delta$ is of a single simplex, the…
We use a numerical method to compute a database of three-point branched covers of the complex projective line of small degree. We report on some interesting features of this data set, including issues of descent.
The diameter of a strongly connected $d$-dimensional simplicial complex is the diameter of its dual graph. We provide a probabilistic proof of the existence of $d$-dimensional simplicial complexes with diameter $ (\frac{1}{d \cdot d!} -…
We prove that random triangulations of high genus contain very large expander subgraphs, answering a question of Benjamini. Our approach relies on new general criteria for arbitrary graphs to contain large expander subgraphs.
We prove general nonlinear large deviation estimates similar to Chatterjee-Dembo's original bounds except that we do not require any second order smoothness. Our approach relies on convex analysis arguments and is valid for a broad class of…
We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial…
We investigate the small deviation probabilities of a class of very smooth stationary Gaussian processes playing an important role in Bayesian statistical inference. Our calculations are based on the appropriate modification of the entropy…
We develop a unified theory to analyze the microcanonical ensembles with several constraints given by unbounded observables. Several interesting phenomena that do not occur in the single constraint case can happen under the multiple…
For statistics of rare events in systems obeying a large-deviation principle, the rate function is a key quantity. When numerically estimating the rate function one is always restricted to finite system sizes. Thus, if the interest is in…
In this article for a finite typed random geometric graph we define the empirical locality distribution, which records the number of nodes of a given type linked to a given number of nodes of each type. We find large deviation principle…