Open research questions in Random Matrices and Applications
178 gap statements mined from Random Matrices and Applications papers in our 4.5M-paper local library, which holds 820 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
What the literature leaves open
The paper suggests that future research could focus on applying the methods and results of the paper to other systems. The paper also suggests that future research could focus on studying the large deviations of the empirical measure of particle systems given by singular Gibbs measures in more detail.
The lower bound for I_N(θ, B_N) uses the unique solution v_N(θ) of equation (14), which requires 1 - 2θb_i + 2θv > 0 for all i. The conditions under which this solution exists and remains uniformly bounded away from the singularity threshold as N varies, particularly for sequences B_N with extreme or clustered entries, requires explicit characterization.
The mathematical description of the quasiparticle description had previously only been realized for two integrable systems. The Toda lattice at thermal equilibrium had not been studied.
The lack of a unified framework for studying iterated and composed CP maps on operator-valued kernels. The need for a transparent realization, where every iterated kernel is the inner product of pushed-forward feature vectors in a single Hilbert space.
The universality of Wigner random matrices has not been fully understood. The comparison of connected correlators of distinct ensembles has not been fully explored.
A new proof of the result that the spectral distributions of the free positive multiplicative Brownian motion form a semigroup with respect to free multiplicative convolution is needed. The properties of the free positive multiplicative Brownian motion and its relation to the free additive convolution of semicircle and uniform distributions need to be further studied.
Free Positive Multiplicative Brownian Motion and the Free Additive Convolution of Semicircle and Uniform Distributions · 2026 · DOIWe expect that exploring this connection and its implications for holography, chaos, and quantum gravity will be a productive direction for future work.
Future research can focus on extending the result to more general β-ensembles. Future research can also explore the applications of the result in various fields, such as physics and engineering.
Large deviations of crowding in finite $β$-ensembles · 2026The paper suggests further research on the application of the grand canonical mixture model to other systems. The paper suggests further research on the improvement of the adaptive time cutting method.
About a nonideal Rayleigh gas mixture model · 2026Future research could focus on applying the result to other statistical mechanics models with continuous symmetry groups. Future research could aim to improve the bound on the Wasserstein distance. Future research could explore the implications of the result for the study of the O(N) model in various contexts.
The Rate of Convergence of the Critical Mean-Field O(N) Magnetization Via Multivariate Nonnormal Stein’s Method · 2026 · DOIThe paper identifies a gap in the existing literature on the O(N) model, namely the lack of a bound on the Wasserstein distance between the finite-volume and limiting distributions. The paper aims to fill this gap by extending a recent multivariate nonnormal approximation theorem to the multivariate O(N) setting.
The Rate of Convergence of the Critical Mean-Field O(N) Magnetization Via Multivariate Nonnormal Stein’s Method · 2026 · DOIThe paper does not provide a comprehensive study of the large-deviation regime for all types of random matrices. The results are limited to the case of random symmetric matrices with independent identically distributed subgaussian entries.
Future research should focus on studying the large-deviation regime for other types of random matrices. The Fourier replacement lemma can be used to study other problems in random matrix theory.
Future research can build on the characterization of the Kirkwood-Dirac representations provided in the paper. Future research can explore the implications of the paper's results for the field of quantum metrology.
What Is Special About the Kirkwood–Dirac Distributions? Only They Produce Natural Conditional Expectations · 2026 · DOIThe paper addresses the need for a characterization of the Kirkwood-Dirac quasiprobability representations.
What Is Special About the Kirkwood–Dirac Distributions? Only They Produce Natural Conditional Expectations · 2026 · DOIFuture research could consider other types of random tensors. The study of eigenpairs of tensors could be extended to other areas, such as quantum gravity and QCD.
The analysis is limited to the projected dice lattice, and it is not clear if the results generalize to other systems.
Parity-dependent double degeneracy and spectral statistics in the projected dice lattice · 2026 · DOIThe analysis applies only to Toda lattices with time horizon s ∈ [0, T] where T ≤ N, but does not address whether the effective velocity approximation remains valid for times T >> N (long-time behavior) or establishes growth rates for the error as a function of T in the Toda lattice dynamics.
The paper notes that the problem of describing sampling sequences in Fock spaces is important and has been studied extensively.
To investigate the incremental universality in other important ensembles, like sparse matrices, band matrices with wide band, block matrices, complex hermitean matrices.
Further study of the properties and applications of the proposed framework. Investigation of the relationships between the proposed framework and other areas of research, such as machine learning and quantum information.
To study the fluctuations of the mod p rank of other types of random matrices. To develop new methods for studying the fluctuations of the mod p rank of triangular matrices.
The paper identifies a gap in the existing literature on large deviation principles for β-ensembles. The paper aims to fill this gap by establishing a large deviation type bound with speed n^2 and a good rate function.
Large deviations of crowding in finite $β$-ensembles · 2026Further study of the interaction Hamiltonian Bn(θ) and its applications. Investigation of the robustness of the technique to environmental effects. Exploration of the technique's potential for quantum-enhanced imaging and continuous-variable QKD protocols.
Detecting nonclassicality in randomly-displaced copies of a squeezed state · 2026The paper identifies a gap in the study of the behavior of tagged particles in a gas mixture. The paper aims to fill this gap by introducing a grand canonical mixture model for a nonideal Rayleigh gas.
About a nonideal Rayleigh gas mixture model · 2026
Most-cited papers in Random Matrices and Applications
- At what sample size do correlations stabilize? · Journal of Research in Personality · 2013 · 1,883 citations
- The Joy of Copulas: Bivariate Distributions with Uniform Marginals · The American Statistician · 1986 · 779 citations
- On the exact distribution of maximally selected rank statistics · Computational Statistics & Data Analysis · 2003 · 576 citations
- On the asymptotic distribution of the Moran I test statistic with applications · Journal of Econometrics · 2001 · 479 citations
- Algorithm 247: Radical-inverse quasi-random point sequence · Communications of the ACM · 1964 · 464 citations
- A general canonical correlation index. · Psychological Bulletin · 1968 · 437 citations
- Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion · Mathematical Social Sciences · 1989 · 415 citations
- The stability–complexity relationship at age 40: a random matrix perspective · Population Ecology · 2015 · 267 citations
- On the diminishing returns of higher-order terms in asymptotic expansions of bias · Economics Letters · 2003 · 263 citations
- On the distributional transform, Sklar's theorem, and the empirical copula process · Journal of Statistical Planning and Inference · 2009 · 208 citations
Most recent work
- 𝑞-Deformed Gaussian unitary ensemble: Spectral moments and genus-type expansions · Transactions of the American Mathematical Society · 2026
- Randomized and Exchangeable Improvements of Markov’s, Chebyshev’s and Chernoff’s Inequalities · Statistical Science · 2026
- S-transform in finite free probability · Advances in Mathematics · 2026
- Limiting laws and consistent estimation criteria for fixed and diverging number of spiked eigenvalues · Journal of the American Statistical Association · 2026
- Moments of the Cramér transform of log-concave probability measures · Journal of Functional Analysis · 2026
- Matrix concentration inequalities and free probability II: Two-sided bounds and applications · Communications of the American Mathematical Society · 2026
- Second-order asymptotic analysis of tail probabilities of bidimensional randomly weighted sums · Science China Mathematics · 2026
- Ties, tails and spectra: On rank-based dependency measures in high dimensions · Stochastic Processes and their Applications · 2026
- Asymptotic analysis of the finite predictor for fractional Gaussian noise · Bernoulli · 2026
- Fourth-moment theorems for sums of multiple integrals · Proceedings of the American Mathematical Society · 2026
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