Probability and Mathematical Physics Seminar

Synopsis:

Renormalization Group (RG) is a powerful concept -- one calculates partition functions by successively integrating out short distance degrees of freedom, until one has an effective Hamiltonian describing only large scale modes. The most famous application is Wilson's use of RG to calculate the critical scaling exponents at the water-vapor phase transition. 

We apply RG concepts to a vastly different class of states, which are far from equilibrium. It is known that a large class of weakly interacting nonlinear systems have states that are spatially homogeneous, time-independent, and scale invariant. Such states, in which mode k has occupation number $n_k = k^{-\gamma}$, go under the name of Kolmogorov-Zakharov states in wave turbulence. Canonical examples are waves on the surface of the ocean, or waves in the nonlinear Schrodinger equation. We compute one loop beta functions in such states, which encode how the effective coupling changes with scale. The beta functions tells us if the spectrum of occupation numbers is steeper or less steep than Kolmogorov-Zakharov scaling.  Depending on the sign of the beta function, nonlinear effects may either cause a minor shift of the state in the IR, or completely change the nature of the state. 

Focusing on nearly marginal interactions (ones in which the strength of the nonlinearity is weakly scale dependent), we construct an analog of Wilson's epsilon expansion and IR fixed points, with epsilon now set by the scaling of the interaction rather than the spacetime dimension. In the language of RG flow, critical balance scaling -- having applications in fields as varied as astrophysics and ocean waves -- corresponds to the state dynamically adjusting itself along the RG flow until the interaction becomes marginal. 

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Notes:

The probability that every vertex in a random orientation of the edges of a given graph has the same in-degree and out-degree is equivalent to counting Eulerian orientations, a problem that is known to be #P-hard in general. This count also appears under the name residual entropy in physical applications, most famously in the study of the behaviour of ice. Using a new tail bound for the cumulant expansion series, we derive an asymptotic formula for graphs of sufficient density. The formula contains the inverse square root of the number of spanning trees, for which we do not have a heuristic explanation. We will also show a strong experimental correlation between the number of spanning trees and the number of Eulerian orientations even for graphs of bounded degree. This leads us to propose a new heuristic for the number of Eulerian orientations which performs much better than previous heuristics for graphs of chemical interest. The talk is based on two recent papers arXiv:2309.15473 and arXiv:2409.04989 joint with B.D. McKay and R.-R. Zhang.

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The 2D two-component Coulomb gas is expected to exhibit an infinite sequence of phase transitions driven by the divergence of 2k-poles, accumulating at the Berezinskii-Kosterlitz-Thouless (BKT) temperature. I will discuss joint work with Sylvia Serfaty in which we provide a rigorous proof of this "dipole transition." Our proof is based on the analysis of dipole pairs and large deviations techniques. I will also discuss multipoles transitions and the BKT transition.

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We prove bounds on statistical distances between high-dimensional exchangeable mixture distributions (which we call permutation mixtures) and their i.i.d. counterparts. Our results are based on a novel method for controlling $\chi^2$ divergences between exchangeable mixtures, which is tighter than the existing methods of moments or cumulants. At a technical level, a key innovation in our proofs is a new Maclaurin-type inequality for elementary symmetric polynomials of variables that sum to zero and an upper bound on permanents of doubly-stochastic positive semidefinite matrices.

Our results imply a de Finetti-style theorem (in the language of Diaconis and Freedman, 1987) and a capacity upper bound for the noisy permutation channel. In particular, when applied to compound decision problems, we tighten the efficiency result of simple decisions in Greenshtein and Ritov (2009), and use empirical Bayes methods to close the gap for competitive distribution estimation in Orlitsky and Suresh (2015).

Based on joint work with Jonathan Niles-Weed (NYU), Yandi Shen (CMU), and Yihong Wu (Yale).

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In this talk I discuss two “driven diffusive systems”, that is models that combine diffusive effects with a forcing mechanism. The first is a diffusion given by the SDE

dXt = F (Xt)dt + dBt

where F is a random drift field and the second is an SPDE given by

∂tu = ∆u + (w · ∇u2) + ∇ · ξ,

where w is a fixed vector, and ξ is space time white noise.

The large-scale behaviour of these models depends highly on the dimension. In dimension 3 and higher, the diffusive effects prevail, while in dimension 1 the nonlinearity dominates. In the scaling critical dimension 2 the behaviour is more subtle and a logarithmic correction to the diffusivity occurs. 

In the talk I will show how to use Fock space analysis to show this behaviour and give an overview of other results in this area. In particular I will contrast the two different universality classes of these models.

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Synopsis:

Evita Nestoridi (11:10 am)

Title: Shuffling via transpositions
Abstract: In their seminal work, Diaconis and Shahshahani proved that shuffling a deck of $n$ cards sufficiently well via random transpositions takes $1/2 n log n$ steps. Their argument was algebraic and relied on the combinatorics of the symmetric group. In this talk, I will focus on a generalization of random transpositions and I will discuss the underlying combinatorics for understanding their mixing behavior and indeed proving cutoff. The talk will be based on joint work with S. Arfaee.

Matthew Kahle (12:10 pm)

Title: Homological percolation in a torus
Abstract: The celebrated Harris–Kesten theorem is that the critical probability for bond percolation in the square lattice is 1/2. It has been a folklore problem for apparently some time to find an appropriate analogue of this theorem in high dimensions.
Duncan, Schweinhart, and I studied "plaquette percolation" in a d-dimensional torus made by identifying opposite sides of a subdivided d-dimensional cube, where k-dimensional cubical cells are inserted independently with probability p. For an appropriate homological definition of "giant cycles", and whenever d=2k, we show that the critical probability is indeed 1/2.

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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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