Gaussian multiplicative chaos through the lens of the 2D Gaussian free field

The aim of this review-style paper is to provide a concise, self-contained and unified presentation of the construction and main properties of Gaussian multiplicative chaos (GMC) measures for log-correlated fields in 2D in the subcritical regime. By considering the case of the 2D Gaussian free field, we review convergence, uniqueness and characterisations of the measures; revisit Kahane's convexity inequalities and existence and scaling of moments; discuss the measurability of the underlying field with respect to the GMC measure and present a KPZ relation for scaling exponents.Comment: 28p, less typos in the 3rd, published version, still no figures, I remain thankful for comment

First passage sets of the 2D continuum Gaussian free field

We introduce the first passage set (FPS) of constant level $-a$ of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path on which the GFF does not go below $-a$. It is, thus, the two-dimensional analogue of the first hitting time of $-a$ by a one-dimensional Brownian motion. We provide an axiomatic characterization of the FPS, a continuum construction using level lines, and study its properties: it is a fractal set of zero Lebesgue measure and Minkowski dimension 2 that is coupled with the GFF $\Phi$ as a local set $A$ so that $\Phi+a$ restricted to $A$ is a positive measure. One of the highlights of this paper is identifying this measure as a Minkowski content measure in the non-integer gauge $r \mapsto \vert\log(r)\vert^{1/2}r^{2}$, by using Gaussian multiplicative chaos theory.Comment: The first version also contained arXiv:1805.09204, which is now a paper on its own; the third version is an all-around improved version ; 42 pages; 8 figures

Sample-level CNN Architectures for Music Auto-tagging Using Raw Waveforms

Recent work has shown that the end-to-end approach using convolutional neural network (CNN) is effective in various types of machine learning tasks. For audio signals, the approach takes raw waveforms as input using an 1-D convolution layer. In this paper, we improve the 1-D CNN architecture for music auto-tagging by adopting building blocks from state-of-the-art image classification models, ResNets and SENets, and adding multi-level feature aggregation to it. We compare different combinations of the modules in building CNN architectures. The results show that they achieve significant improvements over previous state-of-the-art models on the MagnaTagATune dataset and comparable results on Million Song Dataset. Furthermore, we analyze and visualize our model to show how the 1-D CNN operates.Comment: Accepted for publication at ICASSP 201

Intra-Industry Trade in the Baltic Sea Region

The purpose of this paper is to analyse economic integration in the Baltic Sea Region as it has emerged from mid-1990s. More importantly, we seek to assess the quality of integration as conferred by the development of intra-industry trade between the two groups of countries in the Baltic Sea region: Finland, Sweden, Denmark and Germany at the Western coast, and Estonia, Latvia, Lithuania and Poland at the Eastern coast of the Baltic Sea. The analysis of the change in the quality of the traded goods reveals that the economic intergration in the Baltic Sea Region has so far not led to a vast increase of the competitiveness of industry at the relatively less developed Eastern coast of the Baltic Sea. The equalisation of market prices has predominantly taken place in the trade of raw materials, whereas the Eastern countries continue to trade with deficit in the majority of manufactured goods where the equalisation of unit prices has actually taken place. The above seems to support the results of our previous research, in which we have concluded that the economies of the Baltic States and Poland continue to act as lower value-added parts of the cross-border clusters in the Baltic Sea Region. Consequently, if catching up in living standards with the northern and western neighbours is envisioned, much more systematic investment into education and technology is needed in the Baltic States and Poland.Baltic Sea region, intra-industry trade integration

On bounded-type thin local sets of the two-dimensional Gaussian free field

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply-connected domain, and their relation to the conformal loop ensemble CLE(4) and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded-type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the CLE(4) carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of CLE(4)) are in fact measurable functions of the GFF.Comment: 24 pages, to appear in J. Inst. Math. Jussie