Lyapunov exponent, universality and phase transition for products of random matrices

Products of $M$ i.i.d. random matrices of size $N \times N$ are related to classical limit theorems in probability theory ($N=1$ and large $M$), to Lyapunov exponents in dynamical systems (finite $N$ and large $M$), and to universality in random matrix theory (finite $M$ and large $N$). Under the two different limits of $M \to \infty$ and $N \to \infty$, the local singular value statistics display Gaussian and random matrix theory universality, respectively. However, it is unclear what happens if both $M$ and $N$ go to infinity. This problem, proposed by Akemann, Burda, Kieburg \cite{Akemann-Burda-Kieburg14} and Deift \cite{Deift17}, lies at the heart of understanding both kinds of universal limits. In the case of complex Gaussian random matrices, we prove that there exists a crossover phenomenon as the relative ratio of $M$ and $N$ changes from $0$ to $\infty$: sine and Airy kernels from the Gaussian Unitary Ensemble (GUE) when $M/N \to 0$, Gaussian fluctuation when $M/N \to \infty$, and new critical phenomena when $M/N \to \gamma \in (0,\infty)$. Accordingly, we further prove that the largest singular value undergoes a phase transition between the Gaussian and GUE Tracy-Widom distributions.Comment: Therems 1.1 stated in a more intuitive way; proofs extensively revised; convergence in trace norm for the critical and subcritical cases added; 35 pages, 3 figure

Connecting Software Metrics across Versions to Predict Defects

Accurate software defect prediction could help software practitioners allocate test resources to defect-prone modules effectively and efficiently. In the last decades, much effort has been devoted to build accurate defect prediction models, including developing quality defect predictors and modeling techniques. However, current widely used defect predictors such as code metrics and process metrics could not well describe how software modules change over the project evolution, which we believe is important for defect prediction. In order to deal with this problem, in this paper, we propose to use the Historical Version Sequence of Metrics (HVSM) in continuous software versions as defect predictors. Furthermore, we leverage Recurrent Neural Network (RNN), a popular modeling technique, to take HVSM as the input to build software prediction models. The experimental results show that, in most cases, the proposed HVSM-based RNN model has a significantly better effort-aware ranking effectiveness than the commonly used baseline models

Universality for products of random matrices I: Ginibre and truncated unitary cases

Recently, the joint probability density functions of complex eigenvalues for products of independent complex Ginibre matrices have been explicitly derived as determinantal point processes. We express truncated series coming from the correlation kernels as multivariate integrals with singularity and investigate saddle point method for such a type of integrals. As an application, we prove that the eigenvalue correlation functions have the same scaling limits as those of the single complex Ginibre ensemble, both in the bulk and at the edge of the spectrum. We also prove that the similar results hold true for products of independent truncated unitary matrices.Comment: 41 pages; revised upon the suggestions of the anonymous referees; to appear in International Mathematics Research Notices. in IMRN 201

Temperature dependence of circular DNA topological states

Circular double stranded DNA has different topological states which are defined by their linking numbers. Equilibrium distribution of linking numbers can be obtained by closing a linear DNA into a circle by ligase. Using Monte Carlo simulation, we predict the temperature dependence of the linking number distribution of small circular DNAs. Our predictions are based on flexible defect excitations resulted from local melting or unstacking of DNA base pairs. We found that the reduced bending rigidity alone can lead to measurable changes of the variance of linking number distribution of short circular DNAs. If the defect is accompanied by local unwinding, the effect becomes much more prominent. The predictions can be easily investigated in experiments, providing a new method to study the micromechanics of sharply bent DNAs and the thermal stability of specific DNA sequences. Furthermore, the predictions are directly applicable to the studies of binding of DNA distorting proteins that can locally reduce DNA rigidity, form DNA kinks, or introduce local unwinding.Comment: 15 pages in preprint format, 4 figure