Conference Agenda

One-dimensional Calogero-Moser-Sutherland particle models with $N$ particles can be regarded as diffusions on Weyl chambers or alcoves in $\mathbb R^N$ with second order differential operators as generators, which are singular on the boundaries of the state spaces. The most relevant examples are multivariate Bessel and Heckman-Opdam processes which are related to special functions associated with root systems. These models include Dyson's Brownian motions, multivariate Laguerre and Jacobi processes and, for fixed time, $\beta$-Hermite, Laguerre, and Jacobi ensembles. In some cases, they are related to Brownian motions on the classical symmetric spaces.

We review some freezing limits for fixed $N$ when some parameter, an inverse temperature, tends to $\infty$. The limits are normal distributions and, in the process case, Gaussian processes. The parameters of the limits are described in terms of solutions of ordinary differential equations which are frozen versions of the particle diffusions. We discuss connections of these ODES with the zeros of the classical orthogonal polynomials.

The talk is partially based on joint work with Sergio Andraus, Kilian Herrmann, and Jeannette Woerner.

In industry, automated visual surface inspection is a common method for object inspection. Machine learning approaches show promising results for automated defect detection using the acquired images. The performance of such defect detection algorithms increases with the amount and the diversity of training data. However, real data of objects with defects are rarely available in the quantity required for robust machine learning algorithms. Moreover, it is impossible to include the large variety and variability of possible defects since manufacturers try to prevent defect formation.

The lack of data can be overcome by using synthetic training data. That is, we simulate images similar to those obtained by the real surface inspection system by virtually recreating the visual surface inspection environment and the imaging process. This requires a digital twin of the inspected object having the same geometry, material, surface texture and defects. Thus, defect models are necessary to generate synthetic defects and imprint them into the object geometry.

Here, we consider cast metal objects and we focus on defect modeling, more precisely, on two basic defect structures: linear structures such as cracks and flat structures such as surface delaminations. The stochastic models for both defect classes are based on random Voronoi tessellations to reflect the polycrystalline micro-structure of the metal. Models for different defect classes use different features of the tessellations, namely their vertices, edges or cells. Moreover, tessellations with different cell intensities can be used to model the defect structure at different scales and defect variation is guaranteed by using various parameter configurations.

In a classical game two players, Alice and Bob, take turns to play $n$ moves each. Alice starts. For each move each player has two options, 1 and 2. The outcome is determined by the exact sequences of moves played by each player. Prior to the game, a winner is assigned to each of the $2^{2n}$ possible outcomes in an i.i.d. fashion, where $p$ is the probability that Bob is the winner for a given outcome. Then it is known that there exists a value $0<p_{\rm c}<1$ such that the probability that Bob has a winning strategy for large $n$ tends to one if $p>p_{\rm c}$ and to zero if $p<p_{\rm c}$. We study a modification of this game for which the outcome is determined by the exact sequence of moves played by Alice as before, but in the case of Bob all that matters is how often he has played move 1. We show that also in this case, there exists a sharp threshold $p'_{\rm c}$ that determines which player has with large probability a winning strategy in the limit as $n$ tends to infinity.

The hexagonal lattice and its dual, the triangular lattice, are fundamental models for understanding atomic and ring connectivity in carbon structures such as \textit{graphene} and \textit{$(p,q)$--nanotubes}. The chemical and physical properties of these carbon allotropes are closely tied to the average number of closed paths of different lengths $k \in \mathbb{N}_0$ on their respective graph representations, which can be described in terms of their spectra. Since a carbon $(p,q)$--nanotube can be viewed as a graphene sheet rolled up in a manner determined by the \textit{chiral vector} $(p,q)$, our findings are based on the study of \textit{random eigenvalues} for both the hexagonal and triangular lattices presented in \cite{bille23}, as well as previous results on nanotubes \cite{Cotfas00}.

In this talk, we discuss results from \cite{bille24} on the spectral properties of dual infinite $(p,q)$--nanotubes, focusing on the counts of closed paths of length $k \in \mathbb{N}_0$ on these lattices. In particular, for any \textit{chiral vector} $(p,q)$, we show that the sequence of closed path counts forms a moment sequence derived from a functional of two independent uniform distributions. Explicit formulas for key distribution characteristics, including the probability density function and moment generating function, are presented for selected choices of the chiral vector. Moreover, we demonstrate that as the \textit{circumference} of a $(p,q)$--nanotube becomes infinitely large, i.e., as $p+q\rightarrow \infty$, the $(p,q)$--nanotube converges to the hexagonal lattice in terms of the number of closed paths of any given length $k$, indicating weak convergence in the underlying distributions. This approach offers new insights into the spectral behavior of nanotubes and graphene, with practical implications for modeling and analysis in materials science.

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\bibitem{Cotfas00} N. Cotfas. \textit{Random walks on carbon nanotubes and quasicrystals.} Journal of {P}hysics {A}: {M}athematical and {T}heoretical, 33:2917--2927, 2000.

\bibitem{bille23} A. Bille, V. Buchstaber, S. Coste, S. Kuriki and E. Spodarev. \textit{Random eigenvalues of graphenes and the triangulation of plane.} ArXiv preprint No 2306.01462, submitted, 2023. \url{https://arxiv.org/abs/2306.01462}

\bibitem{bille24} A. Bille, V. Buchstaber, P. Ievlev, C. Redenbach, S. Novikov and E. Spodarev. \textit{Random eigenvalues of nanotubes.} ArXiv preprint No 2408.14313, submitted, 2024. \url{https://arxiv.org/abs/2408.14313}.