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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 computation of the probability of rare events is the main purpose of large deviations theory. For instance, in a simple case, one can consider the rare event in which a sum of i.i.d. Bernoulli variables attains a value which is larger than its average. A completely different and much more difficult problem, is the computation of large deviations probability of nonlinear functionals of the Bernoulli variables, e.g. cubic polynomials. A case in which such nonlinear problems arise is, for instance, the study of complex networks. In this talk I will present the behavior of the so-called scaled cumulant generating function of the number of triangles of an Erdӧs-Rényi random graph (dense case). The scaled cumulant generating function is strictly connected with the theory of large deviations since, when it is possible to apply the Gärtner-Ellis theorem, it turns out to be the Legendre transform of the rate function. More precisely, the aim of this talk is twofold. On one hand, to describe a modified version of a known Monte Carlo method, called Cloning algorithm, tailored for approximating the scaled cumulant generating function of an additive observable in the framework of random graphs. One the other hand, keeping the focus on the triangle observable, to present the numerical investigation performed in the region where the analytical expression of such function is not known (replica breaking regime).
The dimer model studies random configurations of perfect matchings (dimer covers) of bipartite planar graphs. Through an associated height function such a configuration is encoded in a random surface. These random surfaces (with a fixed boundary) exhibit limit shape formation: a deterministic limit surface emerges in the macroscopic limit. The imposed boundary condition can have dramatic effect: in certain regions the dimers line up in an ordered fashion (form a frozen facet) and do not look random at all. A prime example of this phenomenon is the arctic circle of domino tilings of the Aztec diamond from 1995. We now have, mostly due to Kenyon et al., a general theory which describes these phenomena in unprecedented detail. The limit shape is described by a convex but singular and degenerate variational problem with a gradient constraint. These features are responsible for facet formation and the appearance of arctic curves. In the talk, I will use the lozenge tiling model (dimer model on the hexagonal lattice) to showcase these issues and address how one can analyse the variational problem.
MSc thesis presentation.
Crossing Probabilities of Multiple Ising Interfaces The planar Ising model is one of the most studied lattice models in statistical physics. Exhibiting a continuous phase transition, it enjoys conformal invariance in the scaling limit, as has been verified recently in celebrated works initiated by S. Smirnov. In this talk, I discuss crossing probabilities of multiple interfaces in the critical Ising model with alternating boundary conditions. In the scaling limit, they are conformally invariant expressions given by so-called pure partition functions of multiple SLE(kappa) with kappa=3. I also describe analogous results for critical percolation and the Gaussian free field. Joint work with Hao Wu (Yau Mathematical Sciences Center, Tsinghua University)
The world we live in is becoming more and more interconnected and huge amounts of data are produced and stored every day by different interrelated entities. This high-throughput generation calls for the development of efficient algorithms to understand and process large and noisy network data. To address this challenging task, we develop a principled approach to summarize (compress) large graphs based on regular partitions, the existence of which was first established in a celebrated result proved by Endre Szemerédi in the mid-1970. A regular partition is defined as a partition of the vertex set into a bounded number of random-like bipartite graphs, called regular pairs. In particular, a regular pair is a highly uniform bipartite graph in which the density of any reasonably sized subgraphs is about the same as the overall density of the bipartite graph. In this talk, I will provide an overview of the regularity lemma and will discuss its potential usefulness in real-world applications.
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.
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