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

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

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

The use of blog to facilitate clinical learning during practicum among undergraduate nursing students: a preliminary study

Blogging has been increasingly used in the education field. This study is the first to explore the use of blogging to facilitate clinical learning for undergraduate nursing students (N=57) with surface or deep study approaches during their clinical practicum in Hong Kong. The findings indicated that blogging is useful in facilitating clinical learning as it fulfills different learning styles and needs of students by promoting information and knowledge sharing, enhancing self-reflection and providing emotional support.postprintThe 6th International Conference on Knowledge Management (ICKM 2009), Hong Kong, 3-4 December 2009. In Proceedings of ICKM, 2009, p. 1-1

A randomized control trial to evaluate the effect of adjuvant selective laser trabeculoplasty versus medication alone in primary open-angle glaucoma: preliminary results

BACKGROUND: The objective of this study was to investigate the effects of adjuvant selective laser trabeculoplasty (SLT) versus medication alone on intraocular pressure (IOP) control, medication use, and quality of life in patients with primary open-angle glaucoma. METHODS: This prospective, randomized control study recruited 41 consecutive primary open-angle glaucoma subjects with medically-controlled IOP /=0.4) or between the two treatment groups (P>/=0.2). CONCLUSION: A single session of adjuvant SLT provided further reductions in IOP and medication without substantial changes in quality of life or medication tolerability at 6 months.published_or_final_versio