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1. Random walk on unipotent matrix groups

   https://doi.org/doi:10.24033/asens.2466  

   Diaconis, Persi; Hough, Robert (May 2021, Annales scientifiques de lÉcole normale supérieure)

   null (Ed.)

   We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure. As a second illustration, the method is used to study walks on the nn uni-upper triangular group with entries taken modulo p. The method allows sharp answers to the behavior of individual coordinates: coordinates immediately above the diagonal require order p^2 steps for randomness, coordinates on the second diagonal require order p steps; coordinates on the kth diagonal require order p^{2/k} steps. 

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2. The local limit theorem on nilpotent Lie groups

   https://doi.org/10.1007/s00440-018-0864-7  

   Hough, Robert (July 2018, Probability Theory and Related Fields)

   A local limit theorem is proven on connected, simply connected nilpotent Lie groups, for a class of generating measures satisfying a moment condition and a condition on the characteristic function of the abelianization. The result extends an earlier local limit theorem of Alexopoulos which treated absolutely continuous measures with a continuous density of compact support, and also extends local limit theorems of Breuillard and Diaconis–Hough which treated general measures on the Heisenberg group. 

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3. Asymptotic shapes for stationary first passage percolation on virtually nilpotent groups

   https://doi.org/10.48550/arXiv.2110.00055  

   Antonio Auffinger; Christian Gorski (September 2021, ArXivorg)

   We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of Benjamini-Tessera and Cantrell-Furman show that scaling limits of such FPP are given by Carnot-Carathéodory metrics on the associated graded nilpotent Lie group. We show a converse, i.e. that for any Cayley graph of a finitely generated nilpotent group, any Carnot-Carathéodory metric on the associated graded nilpotent Lie group is the scaling limit of some FPP with stationary edge weights on that graph. Moreover, for any Cayley graph of any finitely generated virtually nilpotent group, any conjugation-invariant metric is the scaling limit of some FPP with stationary edge weights on that graph. We also show that the conjugation-invariant condition is also a necessary condition in all cases where scaling limits are known to exist. 

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4. Deep variance gamma processes

   https://doi.org/10.1002/sta4.580  

   Berry, Caitlin M.; Kleiber, William (May 2023, Stat)

   Lévy processes are useful tools for analysis and modeling of jump‐diffusion processes. Such processes are commonly used in the financial and physical sciences. One approach to building new Lévy processes is through subordination, or a random time change. In this work, we discuss and examine a type of multiply subordinated Lévy process model that we term a deep variance gamma (DVG) process, including estimation and inspection methods for selecting the appropriate level of subordination given data. We perform an extensive simulation study to identify situations in which different subordination depths are identifiable and provide a rigorous theoretical result detailing the behavior of a DVG process as the levels of subordination tend to infinity. We test the model and estimation approach on a data set of intraday 1‐min cryptocurrency returns and show that our approach outperforms other state‐of‐the‐art subordinated Lévy process models. 

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5. Efficient Approximate Unitary Designs from Random Pauli Rotations

   https://doi.org/10.1109/FOCS61266.2024.00036  

   Haah, Jeongwan; Liu, Yunchao; Tan, Xinyu (October 2024, IEEE)

   We construct random walks on simple Lie groups that quickly converge to the Haar measure for all moments up to order t. The spectral gap of this random walk is shown to be Ω(1/t), which coincides with the best previously known bound for a random walk on the permutation group on {0, 1}^n. This implies that the walk gives an ε-approximate unitary t-design. Our simple proof uses quadratic Casimir operators of Lie algebras. 

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