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1. Combinatorics of the Berezin–Karpelevich integral

   https://doi.org/10.1007/s11005-025-01939-8  

   Novak, Jonathan (May 2025, Letters in Mathematical Physics)

   Abstract The Berezin–Karpelevich integral is a double integral over unitary matrices which plays the role of the Itzykson–Zuber integral in rectangular matrix models. We obtain a topological expansion of the Berezin–Karpelevich integral in terms of monotone Hurwitz numbers and obtain from this certain combinatorial identities. 

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2. A Scalable Hierarchical Semi-Separable Library for Heterogeneous Clusters

   https://doi.org/10.1109/ICPP.2017.60  

   Fernando, Isuru Dilanka; Jayasena, Sanath; Fernando, Milinda; Sundar, Hari (August 2017, 46th International Conference on Parallel Processing (ICPP))

   We present a scalable distributed memory library for generating and computations involving structured dense matrices, such as those produced by boundary integral equation formulations. Such matrices are dense, but have special structure that can be exploited to obtain efficient storage and matrix-vector product evaluations and consequently the fast solution of linear systems. At the core of the methods we use is the observation that off-diagonal matrix blocks of such matrices have a low numerical rank, and that this property can be exploited in a multi-level fashion. In this work we focus on the Hierarchically Semi-Separable (HSS) representation. We present algorithms for building and using HSS representations that are parallelized using MPI and CUDA to leverage state-of-the-art heterogeneous clusters. The efficiency of our methods and implementation is demonstrated on large dense matrices obtained from a boundary integral equation formulation of the Laplace equation with Dirichlet boundary conditions. We demonstrate excellent (linear) scalability on up to 128 GPUs on 128 nodes. Our codes will lay the foundation for fast direct solvers for elliptic problems. 

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3. Spectral form factor of a quantum spin glass

   https://doi.org/10.1007/JHEP09(2022)032  

   Winer, Michael; Barney, Richard; Baldwin, Christopher L.; Galitski, Victor; Swingle, Brian (September 2022, Journal of High Energy Physics)

   A bstract It is widely expected that systems which fully thermalize are chaotic in the sense of exhibiting random-matrix statistics of their energy level spacings, whereas integrable systems exhibit Poissonian statistics. In this paper, we investigate a third class: spin glasses. These systems are partially chaotic but do not achieve full thermalization due to large free energy barriers. We examine the level spacing statistics of a canonical infinite-range quantum spin glass, the quantum p -spherical model, using an analytic path integral approach. We find statistics consistent with a direct sum of independent random matrices, and show that the number of such matrices is equal to the number of distinct metastable configurations — the exponential of the spin glass “complexity” as obtained from the quantum Thouless-Anderson-Palmer equations. We also consider the statistical properties of the complexity itself and identify a set of contributions to the path integral which suggest a Poissonian distribution for the number of metastable configurations. Our results show that level spacing statistics can probe the ergodicity-breaking in quantum spin glasses and provide a way to generalize the notion of spin glass complexity beyond models with a semi-classical limit. 

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4. Parsing the Influence Functional: Harmonic Bath Mapping and Anharmonic Small Matrix Path Integral

   https://doi.org/10.1021/acs.jpclett.4c00908  

   Makri, Nancy (May 2024, The Journal of Physical Chemistry Letters)

   The influence functional (IF) encodes all the information required for calculating dynamical properties of a system in contact with its environment. A direct and simple procedure is introduced for extracting from a few numerical evaluations of the IF, without computing time correlation functions or evaluating integrals, the parameters required for path integral calculations, either within or beyond the harmonic mapping, and for assessing the accuracy of the harmonic bath approximation. Further, the small matrix decomposition of the path integral (SMatPI) is extended to anharmonic environments and the required matrices are constructed directly from the IF. 

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5. Algorithm 1021: SPEX Left LU, Exactly Solving Sparse Linear Systems via a Sparse Left-looking Integer-preserving LU Factorization

   https://doi.org/10.1145/3519024  

   Lourenco, Christopher; Chen, Jinhao; Moreno-Centeno, Erick; Davis, Timothy A. (June 2022, ACM Transactions on Mathematical Software)

   SPEX Left LU is a software package for exactly solving unsymmetric sparse linear systems. As a component of the sparse exact (SPEX) software package, SPEX Left LU can be applied to any input matrix, A , whose entries are integral, rational, or decimal, and provides a solution to the system \( Ax = b \) , which is either exact or accurate to user-specified precision. SPEX Left LU preorders the matrix A with a user-specified fill-reducing ordering and computes a left-looking LU factorization with the special property that each operation used to compute the L and U matrices is integral. Notable additional applications of this package include benchmarking the stability and accuracy of state-of-the-art linear solvers and determining whether singular-to-double-precision matrices are indeed singular. Computationally, this article evaluates the impact of several novel pivoting schemes in exact arithmetic, benchmarks the exact iterative solvers within Linbox, and benchmarks the accuracy of MATLAB sparse backslash. Most importantly, it is shown that SPEX Left LU outperforms the exact iterative solvers in run time on easy instances and in stability as the iterative solver fails on a sizeable subset of the tested (both easy and hard) instances. The SPEX Left LU package is written in ANSI C, comes with a MATLAB interface, and is distributed via GitHub, as a component of the SPEX software package, and as a component of SuiteSparse. 

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