Weyl's law in Liouville quantum gravity

Can you hear the shape of Liouville quantum gravity? We obtain a Weyl law for the eigenvalues of Liouville Brownian motion: the $n$-th eigenvalue grows linearly with $n$, with the proportionality constant given by the Liouville area of the domain and a certain deterministic constant $c_\gamma$ depending on $\gamma \in (0, 2)$. The constant $c_\gamma$, initially a complicated function of Sheffield's quantum cone, can be evaluated explicitly and is strictly greater than the equivalent Riemannian constant. At the heart of the proof we obtain sharp asymptotics of independent interest for the small-time behaviour of the on-diagonal heat kernel. Interestingly, we show that the scaled heat kernel displays nontrivial pointwise fluctuations. Fortunately, at the level of the heat trace these pointwise fluctuations cancel each other which leads to the result. We complement these results by a simulation experiment and discuss a number of conjectures on the spectral geometry of LQG

On the moments of the characteristic polynomial of a Ginibre random matrix

In this article we study the large $N$ asymptotics of complex moments of the absolute value of the characteristic polynomial of a $N\times N$ complex Ginibre random matrix with the characteristic polynomial evaluated at a point in the unit disk. More precisely, we calculate the large $N$ asymptotics of $\mathbb{E}|\det(G_N-z)|^{\gamma}$, where $G_N$ is a $N\times N$ matrix whose entries are i.i.d and distributed as $N^{-1/2}Z$, $Z$ being a standard complex Gaussian, $\Re(\gamma)>-2$, and $|z|<1$. This expectation is proportional to the determinant of a complex moment matrix with a symbol which is supported in the whole complex plane and has a Fisher-Hartwig type of singularity: $\det(\int_\mathbb{C} w^{i}\overline{w}^j |w-z|^\gamma e^{-N|w|^{2}}d^2 w)_{i,j=0}^{N-1}$. We study the asymptotics of this determinant using recent results due to Lee and Yang concerning the asymptotics of orthogonal polynomials with respect to the weight $|w-z|^\gamma e^{-N|w|^2}d^2 w$ along with differential identities familiar from the study of asymptotics of Toeplitz and Hankel determinants with Fisher-Hartwig singularities. To our knowledge, even in the case of one singularity, the asymptotics of the determinant of such a moment matrix whose symbol has support in a two-dimensional set and a Fisher-Hartwig singularity, have been previously unknown.Christian Webb was supported by the Academy of Finland grants 288318 and 308123. Mo Dick Wong is supported by the Croucher Foundation Scholarship and EPSRC grant EP/L016516/1 for his PhD study at Cambridge Centre for Analysis

On Gaussian Multiplicative Chaos

Gaussian multiplicative chaos was first constructed in Kahane's seminal paper in 1985 in an attempt to provide a mathematical foundation for Kolmogorov-Obukhov-Mandelbrot theory of energy dissipation in developed turbulence. It has attracted a lot of attentions from the mathematics community in the last decade, playing a pivotal role in the probabilistic formulation of Liouville conformal field theory, as well as showing up in different branches of mathematics such as analytic number theory where it describes the statistical behaviour of the Riemann zeta function on the critical line. This thesis explores the theory of Gaussian multiplicative chaos in three different directions. We commence with a new connection with random matrix theory, showing that for large Hermitian matrices sampled from the one-cut-regular unitary ensemble, the absolute powers of the characteristic polynomial, when suitably normalised, converge in distribution to multiplicative chaos on the support of the limiting spectral distribution as the size of the matrix goes to infinity, and the limit is independent of the choice of the potential function. This is part of an ongoing programme of establishing Gaussian multiplicative chaos as a universal limit object in probability theory. Next, we consider Gaussian multiplicative chaos in the context of Liouville conformal field theory and study the fusion estimate of the Liouville correlation function. More precisely, we derive the exact asymptotics for the Liouville four-point correlation when two points are merging and express the leading order coefficient in terms of DOZZ constants from the three-point correlation function. Our result is consistent with predictions from conformal bootstrap in theoretical physics, and has a geometric interpretation of surfaces being glued together, as hinted by the bootstrap equation. Finally, we study the right tail of the mass of Gaussian multiplicative chaos and establish a formula for the leading order asymptotics under mild assumptions on the underlying log-correlated Gaussian field. The tail exponent satisfies a universal power-law profile, while the leading order coefficient can be described by the product of two constants, one capturing the dependence on the test set and any non-stationarity, and the other one encoding the universal properties of multiplicative chaos. This may be seen as a first step in understanding the full distributional properties of Gaussian multiplicative chaos.M.D. Wong was supported by the Croucher Foundation Scholarship and EPSRC grant EP/L016516/1 for his PhD study at Cambridge Centre for Analysis

Random Hermitian Matrices and Gaussian Multiplicative Chaos

We prove that when suitably normalized, small enough powers of the absolute value of the characteristic polynomial of random Hermitian matrices, drawn from one-cut regular unitary invariant ensembles, converge in law to Gaussian multiplicative chaos measures. We prove this in the so-called L2-phase of multiplicative chaos. Our main tools are asymptotics of Hankel determinants with Fisher–Hartwig singularities. Using Riemann–Hilbert methods, we prove a rather general Fisher–Hartwig formula for one-cut regular unitary invariant ensembles.N. Berestycki’s work is supported by EPSRC Grants EP/L018896/1 and EP/I03372X/1. M. D. Wong is a PhD student at the Cambridge Centre for Analysis, supported by EPSRC Grant EP/L016516/1. Some of this work was carried out while the first and third authors visited the University of Helsinki, funded in part by EPSRC Grant EP/L018896/1. They also wish to thank the University of Helsinki for its hospitality during this visit. C.Webb wishes to thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the Random Geometry program, during which this project was initiated. C.Webb was supported by the Eemil Aaltonen Foundation grant Stochastic dynamics on large random graphs and Academy of Finland Grants 288318 and 308123

A High Throughput Configurable SDR Detector for Multi-user MIMO Wireless Systems

Spatial division multiplexing (SDM) in MIMO technology significantly increases the spectral efficiency, and hence capacity, of a wireless communication system: it is a core component of the next generation wireless systems, e.g. WiMAX, 3GPP LTE and other OFDM-based communication schemes. Moreover, spatial division multiple access (SDMA) is one of the widely used techniques for sharing the wireless medium between different mobile devices. Sphere detection is a prominent method of simplifying the detection complexity in both SDM and SDMA systems while maintaining BER performance comparable with the optimum maximum-likelihood (ML) detection. On the other hand, with different standards supporting different system parameters, it is crucial for both base station and handset devices to be configurable and seamlessly switch between different modes without the need for separate dedicated hardware units. This challenge emphasizes the need for SDR designs that target the handset devices. In this paper, we propose the architecture and FPGA realization of a configurable sort-free sphere detector, Flex-Sphere, that supports 4, 16, 64-QAM modulations as well as a combination of 2, 3 and 4 antenna/user configuration for handsets. The detector provides a data rate of up to 857.1 Mbps that fits well within the requirements of any of the next generation wireless standards. The algorithmic optimizations employed to produce an FPGA friendly realization are discussed.Xilinx Inc.National Science Foundatio

Kinase and Phosphatase Cross-Talk at the Kinetochore

Multiple kinases and phosphatases act on the kinetochore to control chromosome segregation: Aurora B, Mps1, Bub1, Plk1, Cdk1, PP1, and PP2A-B56, have all been shown to regulate both kinetochore-microtubule attachments and the spindle assembly checkpoint. Given that so many kinases and phosphatases converge onto two key mitotic processes, it is perhaps not surprising to learn that they are, quite literally, entangled in cross-talk. Inhibition of any one of these enzymes produces secondary effects on all the others, which results in a complicated picture that is very difficult to interpret. This review aims to clarify this picture by first collating the direct effects of each enzyme into one overarching schematic of regulation at the Knl1/Mis12/Ndc80 (KMN) network (a major signaling hub at the outer kinetochore). This schematic will then be used to discuss the implications of the cross-talk that connects these enzymes; both in terms of why it may be needed to produce the right type of kinetochore signals and why it nevertheless complicates our interpretations about which enzymes control what processes. Finally, some general experimental approaches will be discussed that could help to characterize kinetochore signaling by dissociating the direct from indirect effect of kinase or phosphatase inhibition in vivo. Together, this review should provide a framework to help understand how a network of kinases and phosphatases cooperate to regulate two key mitotic processes