Probability and Mathematical Physics Seminar

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TBA

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Synopsis:

TBA

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Synopsis:

TBA

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Synopsis:

TBA

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Synopsis:

How is it that natural systems---such as tectonic plates, snow slopes, and financial markets---maintain critical-like states despite no external tuning? I'll go through Bak, Tang, and Wiesenfeld's influential theory of self-organized criticality and then describe our solutions to some long-standing conjectures via the Activated Random Walk model. Joint with Chris Hoffman, Toby Johnson, and Josh Meisel.

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Much of the recent rigorous progress on the classical Ising model was driven by new detailed understanding of its stochastic geometric representations - in particular the random current representation. Motivated by the problem of establishing exponential decay of truncated correlations of the supercritical Ising model in any dimension, Duminil-Copin posed the question in 2016 of determining the (percolative) phase transition of the single random current. By relating the single random current to the loop O(1) model, we prove polynomial lower bounds for path probabilities (and infinite expectation of cluster sizes of 0) for both the single random current and loop O(1) model corresponding to any supercritical Ising model on the hypercubic lattice. 
In this talk, I will gently introduce graphical representations of the Ising model and motivate the theory of percolation followed by a discussion of new results whose surprising proof takes inspiration from the toric code in quantum theory.

Based on joint work with Ulrik Tinggaard Hansen and Boris Kjær: https://arxiv.org/abs/2306.05130

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The old problem of the equilibrium spin systems with quenched disorder remains unsolved and controversial. Different points of view concern whether the equilibrium (Gibbs) states in the low-temperature spin-glass phase of a short-range system decompose into many ordered or ``pure'' states (as suggested by replica symmetry breaking theory (RSB)) or whether instead there are only one or two pure states, related by symmetry (the basis for the scaling-droplet theory). The question does not have much meaning in a finite size system, and so Gibbs states in infinite size are essential. To analyze infinite size, the concept of metastate (introduced by Aizenman, Wehr, Newman, and Stein) is essential. A metastate is a probability distribution on Gibbs states at given disorder, with certain properties. In this talk we describe recent progress in rigorous results on these systems; these results are in accord with, and certainly do not rule out, the predictions of RSB for short-range systems. The rigorous results include a new concept of decomposable metastates, and the result that any metastate can be decomposed into indecomposable metastates.  All results describe what is allowed in a metastate; existence questions of whether non-trivial Gibbs states actually occur will not be considered.

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Tomas Berggren (10-11):

Title: Gaussian free field and discrete Gaussians in periodic dimer models
Abstract: Random dimer models (or equivalently random tiling models) have been extensively studied in mathematics and physics for several decades. A fundamental result is that the associated height function converges, in the large-scale limit, to a deterministic surface known as the limit shape. In this talk, we discuss fluctuations of the height function around the limit shape in the presence of smooth (or gas) regions, specifically on the doubly periodic Aztec diamond dimer model. We show that the height function approximates the sum of two independent components: a Gaussian free field on the multiply connected rough region and a harmonic function with random rough-smooth boundary values. These boundary values are jointly distributed as a discrete Gaussian random vector which maintains a quasi-periodic dependence on $N$. Joint work with Matthew Nicoletti.  

Mateusz Piorkowski (11:15-12:15):

Title: Arctic curves of periodic dimer models and generalized discriminants
Abstract: In this talk I will discuss arctic curves of the k x l-periodic Aztec diamond recently studied by Berggren & Borodin '23 (arXiv:2306.07482). I will show how polynomial equations for these arctic curves can be obtained using theta function of the associated spectral curve. As a corollary we obtain a simple formula for their degree in terms of the number of smooth (or gaseous) and frozen regions. Similar formulas also hold for certain tiling models of the hexagon studied by Bobenko & Bobenko '24 (arXiv:2407.19462).
The key to this result is a generalization of the classical discriminant of a polynomial to the setting of meromorphic sections on compact Riemann surfaces. This talk is based on arXiv:2410.17138.

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We witness many phase transitions in everyday life (eg. ice melting to water). The mathematical approach to these phenomena revolves around the percolation model: given a graph, call each vertex open with probability p independently of the others and look at the subgraph induced by open vertices. Benjamini and Schramm conjectured in 1996 that, at p=1/2, on any planar graph, either there is no infinite connected components or infinitely many.

We prove a stronger version of this conjecture for virtually all planar graphs. We then use this to establish fractal macroscopic behaviour in the loop O(n) model. The latter includes a random discrete Lipschitz surface as a particular case.

Joint work with Matan Harel and Nathan Zelesko.

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Gradient interface models are lattice models from statistical mechanics that arise when describing the microscopic fluctuations of interfaces in nature. While many of their features are believed to be universal, rigorous results are often known only for specific classes of models. I will survey some of these results, and then describe two recent advances: The behavior of the maximum for two-dimensional fields with strictly convex interaction, and scaling limits of the fields for general monotone interactions with quadratic growth.

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Rough calculus is a pathwise extension of the Ito calculus to smooth function(al)s of paths and processes with arbitrary Hölder regularity.
We construct integrals of k-jets along irregular paths with finite p-th variation for any p >1, and derive change of variable formulas for smooth functions of such paths. The case p=2  corresponds to Hans Foellmer's pathwise Ito formula while the case p>2 leads to a "higher order Ito calculus".
The framework is entirely pathwise but applicable to stochastic processes with irregular trajectories, such as (fractional) Brownian motions with arbitrary Hurst exponent. As an application, we show how these results lead to new insights on the transport of measures along rough trajectories.

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