arXiv subjects

Juhan Aru

Publications and source records attributed to Juhan Aru.

27 records · Page 2Linked to original sources

arXiv2018

The first passage sets of the 2D Gaussian free field: convergence and isomorphisms

In a previous article, we introduced 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 along which the GFF is greater than or equal to $-a$. This description can be taken as a definition of the FPS for the metric graph GFF, and it justifies the analogy with the first hitting time of $-a$ by a one-dimensional Brownian motion. In the current article, we prove that the metric graph FPS converges towards the continuum FPS in the Hausdorff metric. This allows us to show that the FPS of the continuum GFF can be represented as a union of clusters of Brownian excursions and Brownian loops, and to prove that Brownian loop soup clusters admit a non-trivial Minkowski content in the gauge $r\mapsto |\log r|^{1/2}r^2$. We also show that certain natural interfaces of the metric graph GFF converge to SLE$_4$ processes.

math.PR↗

arXiv2018

Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics

We study two-valued local sets, $\mathbb{A}_{-a,b}$, of the two-dimensional continuum Gaussian free field (GFF) with zero boundary condition in simply connected domains. Intuitively, $\mathbb{A}_{-a,b}$ is the (random) set of points connected to the boundary by a path on which the values of the GFF remain in $[-a,b]$. For specific choices of the parameters $a, b$ the two-valued sets have the law of the CLE$_4$ carpet, the law of the union of level lines between all pairs of boundary points, or, conjecturally, the law of the interfaces of the scaling limit of XOR-Ising model. Two-valued sets are the closure of the union of countably many SLE$_4$ type of loops, where each loop comes with a label equal to either $-a$ or $b$. One of the main results of this paper describes the connectivity properties of these loops. Roughly, we show that all the loops are disjoint if $a+b \geq 4λ$, and that their intersection graph is connected if $a + b < 4λ$. This also allows us to study the labels (the heights) of the loops. We prove that the labels of the loops are a function of the set $\mathbb{A}_{-a,b}$ if and only if $a\neq b$ and $2λ\leq a+b < 4λ$ and that the labels are independent given the set if and only if $a = b = 2λ$. We also show that the threshold for the level-set percolation in the 2D continuum GFF is $-2λ$. Finally, we discuss the coupling of the labelled CLE$_4$ with the GFF. We characterise this coupling as a specific local set coupling, and show how to approximate these local sets. We further see how in these approximations the labels naturally encode distances to the boundary.

math.PR↗

arXiv2018

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.

math.PR↗

arXiv2017

Liouville measure as a multiplicative cascade via level sets of the Gaussian free field

We provide new constructions of the subcritical and critical Gaussian multiplicative chaos (GMC) measures corresponding to the 2D Gaussian free field (GFF). As a special case we recover E. Aidekon's construction of random measures using nested conformally invariant loop ensembles, and thereby prove his conjecture that certain CLE$_4$ based limiting measures are equal in law to the GMC measures for the GFF. The constructions are based on the theory of local sets of the GFF and build a strong link between multiplicative cascades and GMC measures. This link allows us to directly adapt techniques used for multiplicative cascades to the study of GMC measures of the GFF. As a proof of principle we do this for the so-called Seneta--Heyde rescaling of the critical GMC measure.

math.PR↗

arXiv2017

Two perspectives of the 2D unit area quantum sphere and their equivalence

2D Liouville quantum gravity (LQG) is used as a toy model for 4D quantum gravity and is the theory of world-sheet in string theory. Recently there has been growing interest in studying LQG in the realm of probability theory: David, Kupiainen, Rhodes, Vargas (2014) and Duplantier, Miller, Sheffield (2014) both provide a probabilistic perspective of the LQG on the 2D sphere. In particular, in each of them one may find a definition of the so-called unit area quantum sphere. We examine these two perspectives and prove their equivalence by showing that the respective unit area quantum spheres are the same. This is done by considering a unified limiting procedure for defining both objects.

math.PR↗

arXiv2017

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.

math.PR↗

arXiv2015

Multiagent Cooperation and Competition with Deep Reinforcement Learning

Multiagent systems appear in most social, economical, and political situations. In the present work we extend the Deep Q-Learning Network architecture proposed by Google DeepMind to multiagent environments and investigate how two agents controlled by independent Deep Q-Networks interact in the classic videogame Pong. By manipulating the classical rewarding scheme of Pong we demonstrate how competitive and collaborative behaviors emerge. Competitive agents learn to play and score efficiently. Agents trained under collaborative rewarding schemes find an optimal strategy to keep the ball in the game as long as possible. We also describe the progression from competitive to collaborative behavior. The present work demonstrates that Deep Q-Networks can become a practical tool for studying the decentralized learning of multiagent systems living in highly complex environments.

cs.AI↗

arXiv2014

Untangling cross-frequency coupling in neuroscience

Cross-frequency coupling (CFC) has been proposed to coordinate neural dynamics across spatial and temporal scales. Despite its potential relevance for understanding healthy and pathological brain function, the standard CFC analysis and physiological interpretation come with fundamental problems. For example, apparent CFC can appear because of spectral correlations due to common non-stationarities that may arise in the total absence of interactions between neural frequency components. To provide a road map towards an improved mechanistic understanding of CFC, we organize the available and potential novel statistical/modeling approaches according to their biophysical interpretability. While we do not provide solutions for all the problems described, we provide a list of practical recommendations to avoid common errors and to enhance the interpretability of CFC analysis.

q-bio.NC↗

arXiv2014

KPZ relation does not hold for the level lines and the SLE$_κ$ flow lines of the Gaussian free field

In this paper we mingle the Gaussian free field, the Schramm-Loewner evolution and the KPZ relation in a natural way, shedding new light on all of them. Our principal result shows that the level lines and the SLE$_κ$ flow lines of the Gaussian free field do not satisfy the usual KPZ relation. In order to prove this, we have to make a technical detour: by a careful study of a certain diffusion process, we provide exact estimates of the exponential moments of winding of chordal SLE curves conditioned to pass nearby a fixed point. This extends previous results on winding of SLE curves by Schramm.

math.PR↗