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

Analyzing possible pitfalls of cross-frequency analysis : poster presentation from Twentieth Annual Computational Neuroscience Meeting CNS*2011 Stockholm, Sweden, 23 - 28 July 2011

Poster presentation from Twentieth Annual Computational Neuroscience Meeting: CNS*2011 Stockholm, Sweden. 23-28 July 2011. One of the central questions in neuroscience is how neural activity is organized across different spatial and temporal scales. As larger populations oscillate and synchronize at lower frequencies and smaller ensembles are active at higher frequencies, a cross-frequency coupling would facilitate flexible coordination of neural activity simultaneously in time and space. Although various experiments have revealed amplitude-to-amplitude and phase-to-phase coupling, the most common and most celebrated result is that the phase of the lower frequency component modulates the amplitude of the higher frequency component. Over the recent 5 years the amount of experimental works finding such phase-amplitude coupling in LFP, ECoG, EEG and MEG has been tremendous (summarized in [1]). We suggest that although the mechanism of cross-frequency-coupling (CFC) is theoretically very tempting, the current analysis methods might overestimate any physiological CFC actually evident in the signals of LFP, ECoG, EEG and MEG. In particular, we point out three conceptual problems in assessing the components and their correlations of a time series. Although we focus on phase-amplitude coupling, most of our argument is relevant for any type of coupling. 1) The first conceptual problem is related to isolating physiological frequency components of the recorded signal. The key point is to notice that there are many different mathematical representations for a time series but the physical interpretation we make out of them is dependent on the choice of the components to be analyzed. In particular, when one isolates the components by Fourier-representation based filtering, it is the width of the filtering bands what defines what we consider as our components and how their power or group phase change in time. We will discuss clear cut examples where the interpretation of the existence of CFC depends on the width of the filtering process. 2) A second problem deals with the origin of spectral correlations as detected by current cross-frequency analysis. It is known that non-stationarities are associated with spectral correlations in the Fourier space. Therefore, there are two possibilities regarding the interpretation of any observed CFC. One scenario is that basic neuronal mechanisms indeed generate an interaction across different time scales (or frequencies) resulting in processes with non-stationary features. The other and problematic possibility is that unspecific non-stationarities can also be associated with spectral correlations which in turn will be detected by cross frequency measures even if physiologically there is no causal interaction between the frequencies. 3) We discuss on the role of non-linearities as generators of cross frequency interactions. As an example we performed a phase-amplitude coupling analysis of two nonlinearly related signals: atmospheric noise and the square of it (Figure 1) observing an enhancement of phase-amplitude coupling in the second signal while no pattern is observed in the first. Finally, we discuss some minimal conditions need to be tested to solve some of the ambiguities here noted. In summary, we simply want to point out that finding a significant cross frequency pattern does not always have to imply that there indeed is physiological cross frequency interaction in the brain

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

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

Temperature Measurement during Thermonuclear X-ray Bursts with BeppoSAX

We have carried out a study of temperature evolution during thermonuclear bursts in LMXBs using broad band data from two instruments onboard BeppoSAX, the MECS and the PDS. However, instead of applying the standard technique of time resolved spectroscopy, we have determined the temperature in small time intervals using the ratio of count rates in the two instruments assuming a blackbody nature of burst emission and different interstellar absorption for different sources. Data from a total of twelve observations of six sources were analysed during which 22 bursts were detected. We have obtained temperatures as high as ~3.0 keV, even when there is no evidence of photospheric radius expansion. These high temperatures were observed in the sources within different broadband spectral states (soft and hard).Comment: To appear in New Astronom

Balancing sums of random vectors

We study a higher-dimensional 'balls-into-bins' problem. An infinite sequence of i.i.d. random vectors is revealed to us one vector at a time, and we are required to partition these vectors into a fixed number of bins in such a way as to keep the sums of the vectors in the different bins close together; how close can we keep these sums almost surely? This question, our primary focus in this paper, is closely related to the classical problem of partitioning a sequence of vectors into balanced subsequences, in addition to having applications to some problems in computer science.Comment: 17 pages, Discrete Analysi