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Aalto Stochastics and Statistics Seminar is organized by Pauliina Ilmonen, Kalle Kytölä, and Lasse Leskelä. Feel free to contact one of us if you are interested in giving a talk.
The scaling limit of the 2D critical Ising model is expected to exhibit conformal invariance, which has been proved in the case of spin (Chelkak, Hongler and Izyurov 2015) and energy (local 2-point) correlations (Hongler and Smirnov 2013). We extend these results by giving a conformally covariant description for the local n-point correlations in the scaling limit. Then we will go on to discuss preliminary results and ongoing work in the near-critical (scaling towards criticality) setting and their significance from the viewpoint of Conformal Field Theory. Based on joint work with R. Gheissari (first part) and C. Hongler.
A major conjecture in two-dimensional statistical mechanics is that scaling limits of lattice models are described by conformally invariant quantum field theories. The main ingredient of conformal field theories is a Virasoro algebra representation on local fields. In two lattice models, the critical Ising model and the discrete Gaussian free field, we find this exact structure even before passing to the scaling limit. The result is joint work with Clément Hongler (EPFL) and Fredrik Viklund (KTH).
I will discuss some recent work concerning rigidity of eigenvalues of a random Hermitian matrix: that is, how much can the eigenvalues of a large random Hermitian matrix fluctuate around certain deterministic quantities.
One of the standard axiomatic approaches to conformal field theory involves infinite-dimensional moduli spaces of Riemann surfaces. The rigorous definition and study of these moduli spaces requires complex analysis and geometry, quasiconformal Teichmueller theory, functional analysis etc. I will outline the connection between these topics and state some recent results.
In statistical mechanics, one often studies a random model on a fine-mesh lattice approximating a continuum domain. One descriptive feature of such models are interface curves, such as boundaries of magnetization clusters in a ferromagnetism model. Consider a situation where boundary conditions, for instance alternating +/- magnetizations on the boundary of the ferromagnet, force the existence of multiple macroscopic chordal interfaces. We derive criteria, valid for various random models, that guarantee the existence of a weak limit of such chordal interfaces as the lattice mesh turns finer.
I will explain a method that has recently appeared in metric geometry and has shown to be an effective technique to compactify metric spaces. Afterwards, I will present a complete description of the metric compactification of the classical Banach spaces Lp in finite and infinite dimensions.
I will discuss how discrete complex analysis and orthogonal polynomials can be used to study the spin correlation functions of the two dimensional Ising model.
A self avoiding walk (SAW) is an injective walk in a lattice embedded in an Euclidean space. A random SAW of length n is a uniformly distributed random variable on all self avoiding walks of length n starting from origin. In this talk, we are mainly interested in random SAWs in a d-dimensional qubic lattice which are restricted to the upper half plane. These are called self avoiding half space random walks (SAHSW). I will present necessary tools to show the existence of an infinite random SAHSW which is defined as the limit of random SAHSWs of length n, as n tends to infinity.
In this talks we characterize topological duals of Frechet spaces of stochastic processes. This is done by analyzing the optional projection on spaces of cadlag processes whose pathwise supremum norm belongs to a given Frechet space of random variables. We employ functional analytic arguments that unify various results in the duality theory of stochastic processes and also yields new ones of both practical and theoretical interest. In particular, we find an explicit characterization of dual of the Banach space of adapted cadlag processes of class (D). When specialized to regular processes, we obtain a simple proof of a result of Bismut on projections of continuous processes.
Depth has become a quite popular concept in functional data analysis. In the talk we discuss its general framework. We show that most known functional depths can be classified into few groups, within which they share similar theoretical properties. We focus on uniform consistency results for the sample versions of these functionals, and demonstrate that some well-known approaches to depth assessment are hardly theoretically adequate.
We extend a classic method of independent component analysis, the fourth order blind identification (FOBI), to vector-valued functional data. The use of multivariate instead of univariate functions allows for natural definitions for both the marginals of a random function and their mutual independence. Our model assumes that the observed functions are mixtures of latent independent functions residing in suitable Hilbert spaces, mixed with a bounded linear operator from the product space to itself. To enable the inversion of the covariance operator we make the assumption that the dependency between the mixed component functions lies in a finite-dimensional subspace. In this subspace we define fourth cross-cumulant operators and use them to construct a novel Fisher consistent method for solving the independent component problem for vector-valued functions. Finally, both simulations and an application to hand gesture data set are used to demonstrate the advantages of the proposed method over its closest competitors.
Most of the functional depth approaches presented in the literature are solely interested in the -pointwise- centrality of the observations as a measure of (global) centrality. As a result, they are missing some important features inherent to functional data such as variation in shape, roughness or range. Thus, due to the richness of functional data, we opt to talk about typicality rather than centrality of an observation. We discuss assessing typicality of functional observations. Moreover, we provide a new concept of depth for functional data. It is based on a new multivariate Pareto depth applied after mapping the functional observations to a vector of statistics of interest. These quantities allow incorporating the inherent features of the distribution, such as shape or roughness. In particular, in contrast to most existing functional depths, the method is not limited to centrality only. Properties of the new depth are explored and the benefits of a flexible choice of features are illustrated.
We propose halfspace depth concepts for scatter, concentration and shape matrices. For scatter matrices, our concept extends the one from Chen, Gao and Ren (2015) to the non-centered case, and is in the same spirit as the one in Zhang (2002). Rather than focusing, as in these earlier works, on deepest scatter matrices, we thoroughly investigate the properties of the proposed depth and of the corresponding depth regions. We do so under minimal assumptions and, in particular, we do not restrict to elliptical distributions nor to absolutely continuous distributions. Interestingly, fully understanding scatter halfspace depth requires considering different geometries/topologies on the space of scatter matrices. We also discuss, in the spirit of Zuo and Serfling (2000), the structural properties a scatter depth should satisfy, and investigate whether or not these are met by the proposed depth. As mentioned above, companion concepts of depth for concentration matrices and shape matrices are also proposed and studied.
Many commonly used drugs target intracellular signaling pathways that have importance in carcinogenesis and progression of cancer. One example is the mevalonate pathway which is targeted by cholesterol-lowering statin drugs. Examining possible anticancer effects of drug groups established for other indications requires comprehensive multidisciplinary approach including epidemiological studies to explore associations with cancer risk and prognosis, experimental laboratory studies to elucidate possible anticancer mechanisms and ultimately clinical trials to test for clinical benefit. This lecture describes such project testing statins' effects against prostate cancer.
The first independent component analysis estimators were designed for i.i.d. data, and they assume that at most one of the independent components has Gaussian density. Afterwards, methods have been developed for data with dependencies between the observations, including for example time series and spatial data. Those methods utilize solely or partially the structure of the data. For the most general case, graph data, Blochl et al. introduced an ICA estimator called GraDe (graph decorrelation) which uses approximate joint diagonalization of graph-autocorrelation matrices. GraDe contains the best-known time series and spatial methods as special cases. The structure of the graph is given in an adjacency matrix which is to be known or estimated prior to performing the ICA task. As one example in our paper on the effects of adjacency matrix estimation errors in graph signal processing, we study how the GraDe method is affected by misspecification of the adjacency matrix, using both theoretical and simulation results.
Let $X_1$ and $N$ be non-negative integer valued power law random variables. For a randomly stopped sum $S_N=X_1+\cdots+X_N$ of independent and identically distributed copies of $X_1$ we establish a first order asymptotics of the local probabilities $P(S_N=t)$ as $t\to+\infty$. Using this result we show the $k^{-\delta}$, $0\le \delta\le 1$ scaling of the local clustering coefficient (of a randomly selected vertex of degree $k$) in a power law affiliation network. http://arxiv.org/abs/1801.01035
A simple mathematical model for information spreading is a complete graph where everyone knows everyone and everyone relays messages at random time instants to randomly chosen neighbours independently of each other. In this talk we will consider information spreading in the large configuration model, which is a more advanced model. One of the key quantity analysing the rumour spreading speed is the broadcast time, the time when the rumour has reached the entire population. We will also discuss the broadcast time taking into account the passive and active users.
A physics principle asserts that the scaling limit of a critical lattice model, as increasingly dense lattices approximate a continuum domain, is described by a conformal field theory (CFT). The aim of this talk is to prove rigorously conformal invariance properties in the scaling limit of the uniform spanning tree (UST) of a planar graph, i.e., a tree subgraph that covers all the vertices of the original graph, chosen uniformly at random. In a spanning tree, any two vertices are connected by a unique path, called a branch. We study multiple simultaneous UST boundary-to-boundary branches between given boundary vertices, as well as the boundary visits of a single such branch. The related probabilities have conformally invariant scaling limits, and solve partial differential equations as predicted by CFT. As a collection of curves, such multiple simultaneous branches converge weakly to a conformally invariant law, called the local multiple SLE(2). These are among the first verifications of third-order PDEs of CFT, as well as convergence results to multiple SLE.
The horofunction compactification is the result of making any metric space into a compact topological space by only using the metric. The elements of this compactification have been recently shown to be very useful in the study of limit theorems for deterministic and random dynamical systems. In this talk I will give a complete description of the horofunction compactification of the classical lp spaces. I will also consider Hilbert's projective metric on the standard cone of positive real vectors, and describe its horofunction compactification. The latter will be used to give a new proof of the well-known Perron theorem.
For a long time network science concentrated on static graphs as representations of networked systems. This abstraction was used to analyse shortest path lengths, spreading of disease or information in networks, and many other things. Often the underlying assumption behind such analysis was that the activation of nodes and links is controlled by homogeneous Poisson processes. More recently there has been growing interest in analysing 'temporal networks' where the link activation times are determined directly from data. In this talk I will illustrate how this approach can be used to analyse a large communication network with hundreds of millions of link activation events. I will focus on how 'reference models', in which the data is shuffled in various ways, can be used for this data and how they are used in the literature on temporal networks in general. I will also discuss the challenges in the literature that are caused by the sudden increase in use of such shuffling methods of temporal network.
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