The tip multifractal spectrum of a 2-dimensional curve is one way to describe the behavior of the uniformizing conformal map of the complement near the tip. We give the tip multifractal spectrum for a Schramm–Loewner evolution (SLE) curve, we prove that the spectrum is valid with probability 1, and we give applications to the scaling of harmonic measure at the tip.

We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\le N \le M \le dN$ for some constant $d\ge 1$. Let $X$ be an $M\times n$ random matrix with independent and identically distributed entries, which have zero mean, unit variance and arbitrarily high moments. Let $T$ be an $N\times M$ deterministic matrix with comparable singular values $c\le s_{N}(T) \le s_{1}(T) \le c^{-1}$ for some constant $c>0$. Let $A$ be an $N\times n$ deterministic matrix with $\|A\|=O(\sqrt{N})$. Then we prove that for any $\epsilon>0$, the smallest singular value of $TX-A$ is larger than $N^{-\epsilon}(\sqrt{N}-\sqrt{n-1})$ with high probability. If we assume further the entries of $X$ have subgaussian decay, then the smallest singular value of $TX-A$ is at least of the order $\sqrt{N}-\sqrt{n-1}$ with high probability, which is an essentially optimal estimate.

We develop a generic game platform that can be used to model various real-world systems with multiple intelligent cloud-computing pools and parallel-queues for resources-competing users. Inside the platform, the software structure is modelled as Blockchain. All the users are associated with Big Data arrival streams whose random dynamics is modelled by triply stochastic renewal reward processes (TSRRPs). Each user may be served simultaneously by multiple pools while each pool with parallel- servers may also serve multi-users at the same time via smart policies in the Blockchain, e.g. a Nash equilibrium point myopically at each fixed time to a game-theoretic scheduling problem. To illustrate the effectiveness of our game platform, we model the performance measures of its internal data flow dynamics (queue length and workload processes) as reflecting diffusion with regime-switchings (RDRSs) under our scheduling policies. By RDRS models, we can prove our myopic game-theoretic policy to be an asymptotic Pareto minimal-dual-cost Nash equilibrium one globally over the whole time horizon to a randomly evolving dynamic game problem. Iterative schemes for simulating our multi-dimensional RDRS models are also developed with the support of numerical comparisons.

It is well known that the spectral measure of eigenvalues of a rescaled square non-Hermitian random matrix with independent entries satises the circular law. We consider the product $T X$, where $T$ is a deterministic $N\times M$ matrix and $X$ is a random $M\times  N$ matrix with independent entries having zero mean and variance $(N \wedge M) ^{-1}$. We prove a general local circular law for the empirical spectral distribution (ESD) of $TX$ at any point $z$ away from the unit circle under the assumptions that$ N \sim M$, and the matrix entries $X_{ij} have suciently high moments. More precisely, if $z$ satisfes $| |z|
- 1| \ge \tau$  for arbitrarily small $\tau > 0, the ESD of $TX$ converges to $\tilde{X}_D (z) d A(z)$, where $\tilde{X}_D$ is a rotation-invariant function determined by the singular valuau of $T$ and $dA$ denotes the Lebesgue measure on $\mathbb{C}$. The local circular law is valid around $z$ up to scale $(N \wedge M) ^{-1/4+\epsilon}$ for any  $\epsilon >0$. Moreover, if $|z| >1$ or the matrix entries of $X$ have vanishing third moments, the local circular law is valid around $z$ up to scale $(N \wedge M) ^{-1/2+\epsilon}$ for any  $\epsilon >0$.

The {\it{eigenvector empirical spectral distribution}} (VESD) is a useful tool in studying the limiting behavior of eigenvalues and eigenvectors of covariance matrices. In this paper, we study the convergence rate of the VESD of sample covariance matrices to the deformed Mar{\v c}enko-Pastur (MP) distribution. Consider sample covariance matrices of the form $\Sigma^{1/2} X X^* \Sigma^{1/2}$, where $X=(x_{ij})$ is an $M\times N$ random matrix whose entries are independent random variables with mean zero and variance $N^{-1}$, and $\Sigma$ is a deterministic positive-definite matrix. We prove that the Kolmogorov distance between the {\it{expected VESD}} and the deformed MP distribution is bounded by $N^{-1+\epsilon}$ for any fixed $\epsilon>0$, provided that the entries $\sqrt{N}x_{ij}$ have uniformly bounded 6th moments and $|N/M-1|\ge \tau$ for some constant $\tau>0$. This result improves the previous one obtained in [44], which gave the convergence rate $O(N^{-1/2})$ assuming $i.i.d.$ $X$ entries, bounded 10th moment, $\Sigma=I$ and $M<N$. Moreover, we also prove that under the finite $8$th moment assumption, the convergence rate of the VESD is $O(N^{-1/2+\epsilon})$ almost surely for any fixed $\epsilon>0$, which improves the previous bound $N^{-1/4+\epsilon}$ in [44].