L6

In this talk, I will discuss the challenges of solving string theory through the two-dimensional quantum field theory on the string worldsheet. Focusing on the spectrum of superstrings in AdS3, I will explain how conformal field theory techniques yield exact results for special backgrounds, and how methods from integrable models allow us to go beyond these cases.

TBA

Let $W=W^*$ be a large random matrix. For a broad range of ensembles it is known that the resolvent $G(z):=(W-z)^{-1}$ concentrates around a deterministic quantity as long as $|\Im z|$ is larger than the local eigenvalue spacing of $W$. This is the so-called single-resolvent local law. Multi-resolvent local laws extend this framework by showing that products of several resolvents also concentrate. In this talk, I will discuss two phenomena that provide additional information beyond the standard local laws: decorrelation in energy space and gains from taking expectations. I will also present several recent applications of these results, which would not be accessible via standard local laws. These applications include non-Gaussian corrections to fluctuations in local laws, law of fractional logarithm for Wigner minor process and hyperuniformity of eigenvalues of non-Hermitian random matrices.  The talk is based on several recent joint works with Z. Bao, G. Cipolloni, L. Erd{\H o}s and J. Henheik.

Wireless communication systems with many antennas and users naturally give rise to large random channel matrices. In one-bit precoding, the transmitted signal is restricted to binary values and is designed using the channel matrix. It is then multiplied by the same channel matrix during transmission. This dependence, together with the entrywise sign nonlinearity, makes the performance difficult to characterize. Focusing on the standard i.i.d. Gaussian channel model, this talk studies the large-system behaviors of two classes of one-bit precoding schemes. For linear-quantized precoding, we use a recursive representation of Haar matrices, known as Householder Dice, to construct an asymptotic scalar signal-plus-independent-Gaussian-noise model.  We next consider nonlinear symbol-level precoding, focusing on a scheme based on convex relaxation followed by one-bit quantization. Using approximate message passing and state evolution, we characterize the limiting empirical laws of the relaxed solution and of the received signal–symbol pairs. These results provide explicit performance predictions and guidance for precoder design.

tbc