10th World Congress in Probability and Statistics
Invited Session (live Q&A at Track 3, 10:30PM KST)
Invited 13
Critical Phenomena in Statistical Mechanics Models (Organizer: Akira Sakai)
Conference
10:30 PM
—
11:00 PM KST
Local
Jul 19 Mon, 8:30 AM
—
9:00 AM CDT
Recent results for critical lattice models in high dimensions
Mark Holmes (University of Melbourne)
9
We’ll discuss recent results concerning the limiting behaviour of critical lattice models (e.g. lattice trees and the voter model) in high dimensions. In particular (i) results with Ed Perkins on the scaling limit of the range (the set of vertices ever visited/occupied by the model), and (ii) results with Cabezas, Fribergh, and Perkins on weak convergence of the historical processes and of random walks on lattice trees.
Near-critical avalanches in 2D frozen percolation and forest fires
Pierre Nolin (City University of Hong Kong)
8
We discuss two closely related processes in two dimensions: frozen percolation, where connected components of occupied vertices freeze (they stop growing) as soon as they reach a given size, and forest fire processes, where connected components are hit by lightning (and thus become entirely vacant) at a very small rate. When the density of occupied sites approaches the critical threshold for Bernoulli percolation, both processes display a striking phenomenon: the appearance of what we call "near-critical avalanches”. We study these avalanches, all the way up to the natural characteristic scale of each model, which constitutes an important step toward understanding the self-organized critical behavior of such processes. In the case of forest fires, it is crucial to analyze the effect of fires on the connectivity of the forest. For this purpose, a key tool is a percolation model where regions ("impurities") are removed from the lattice, in an independent fashion. The macroscopic behavior of this process is quite subtle, since the impurities are not only microscopic, but also allowed to be mesoscopic.
This talk is based on joint works with Rob van den Berg (CWI and VU, Amsterdam) and with Wai-Kit Lam (University of Minnesota).
This talk is based on joint works with Rob van den Berg (CWI and VU, Amsterdam) and with Wai-Kit Lam (University of Minnesota).
Quenched and annealed Ising models on random graphs
Cristian Giardinà (Modena & Reggio Emilia University)
6
The ferromagnetic Ising model on a lattice is a paradigmatic model of statistical physics used to study phase transitions in lattice systems. In this talk, I shall consider the setting where the regular spatial structure of a lattice is replaced by a random graph, which is often used to model complex networks. I shall treat both the case where the graph is essentially frozen (quenched setting) and the case where instead it is rapidly changing (annealed setting). I shall prove that quenched and annealed may have different critical temperatures, provided the graph degrees are allowed to fluctuate. I shall also discuss how universal results (law of large numbers, central limit theorems, critical exponents) are affected by the disorder in the spatial structure.
The picture that I will present emerges from several joint works, involving V.H. Can, S. Dommers, C. Giberti, R.van der Hofstad and M.L.Prioriello.
The picture that I will present emerges from several joint works, involving V.H. Can, S. Dommers, C. Giberti, R.van der Hofstad and M.L.Prioriello.
Q&A for Invited Session 13
0
This talk does not have an abstract.
Session Chair
Akira Sakai (Hokkaido University)
Invited 15
Privacy (Organizer: Angelika Rohde)
Conference
10:30 PM
—
11:00 PM KST
Local
Jul 19 Mon, 8:30 AM
—
9:00 AM CDT
The Right Complexity Measure in Locally Private Estimation: It is not the Fisher Information
John Duchi (Stanford University)
2
We identify fundamental tradeoffs between statistical utility and privacy under local models of privacy in which data is kept private even from the statistician, providing instance-specific bounds for private estimation and learning problems by developing the local minimax risk. In contrast to approaches based on worst-case (minimax) error, which are conservative, this allows us to evaluate the difficulty of individual problem instances and delineate the possibilities for adaptation in private estimation and inference. Our main results show that the local modulus of continuity of the estimand with respect to the variation distance—as opposed to the Hellinger distance central to classical statistics—characterizes rates of convergence under locally private estimation for many notions of privacy, including differential privacy and its relaxations. As a consequence of these results, we identify an alternative to the Fisher information for private estimation, giving a more nuanced understanding of the challenges of adaptivity and optimality.
Sequentially interactive versus non-interactive local differential privacy: estimating the quadratic functional
Lukas Steinberger (University of Vienna)
2
We develop minimax rate optimal locally differentially private procedures for estimating the integrated square of the data generating density. A sequentially interactive two-step procedure is found to outperform the best possible non-interactive method even in terms of convergence rate. This is in stark contrast to many other private estimation problems (e.g., those where the estimand is a linear functional of the data generating distribution) where it is known that sequential interaction between data owners can not lead to faster rates of estimation than that of an optimal non-interactive method.
Gaussian differential privacy
Weijie Su (University of Pennsylvania)
3
Privacy-preserving data analysis has been put on a firm mathematical foundation since the introduction of differential privacy (DP) in 2006. This privacy definition, however, has some well-known weaknesses: notably, it does not tightly handle composition. In this talk, we propose a relaxation of DP that we term "f-DP", which has a number of appealing properties and avoids some of the difficulties associated with prior relaxations. First, f-DP preserves the hypothesis testing interpretation of differential privacy, which makes its guarantees easily interpretable. It allows for lossless reasoning about composition and post-processing, and notably, a direct way to analyze privacy amplification by subsampling. We define a canonical single-parameter family of definitions within our class that is termed "Gaussian Differential Privacy", based on hypothesis testing of two shifted normal distributions. We prove that this family is focal to f-DP by introducing a central limit theorem, which shows that the privacy guarantees of any hypothesis-testing based definition of privacy (including differential privacy) converge to Gaussian differential privacy in the limit under composition. This central limit theorem also gives a tractable analysis tool. We demonstrate the use of the tools we develop by giving an improved analysis of the privacy guarantees of noisy stochastic gradient descent.
This is joint work with Jinshuo Dong and Aaron Roth.
This is joint work with Jinshuo Dong and Aaron Roth.
Q&A for Invited Session 15
0
This talk does not have an abstract.
Session Chair
Angelika Rohde (University of Freiburg)
Invited 24
Random Planar Geometries (Organizer: Nina Holden)
Conference
10:30 PM
—
11:00 PM KST
Local
Jul 19 Mon, 8:30 AM
—
9:00 AM CDT
Markovian infinite triangulations
Thomas Budzinski (École normale supérieure de Lyon)
5
We say that a random infinite planar triangulation T is Markovian if for any small triangulation t with boundaries, the probability to observe t around the root of T only depends on the boundaries and the total size of t. Such a property can be expected from the local limits of many natural models of random maps. An important example is the UIPT of Angel and Schramm, which is the local limit of large uniform triangulations of the sphere. We will classify completely infinite Markovian planar triangulations, without any assumption on the number of ends. In particular, we will see that there is (almost) no model of multi-ended Markovian triangulation. As an application, we will prove, without to rely on enumerative combinatorics, that the convergence of uniform triangulations to the UIPT is robust under certain perturbations. One example of such a perturbation is to consider random maps with prescribed face degrees where almost all faces are triangles.
Rotational invariance in planar FK-percolation
Ioan Manolescu (Université de Fribourg)
4
We prove the asymptotic rotational invariance of the critical FK-percolation model on the square lattice with any cluster-weight between 1 and 4. These models are expected to exhibit conformally invariant scaling limits that depend on the cluster weight, thus covering a continuum of universality classes. The rotation invariance of the scaling limit is a strong indication of the wider conformal invariance, and may indeed serve as a stepping stone to the latter.
Our result is obtained via a universality theorem for FK-percolation on certain isoradial lattices. This in turn is proved via the star-triangle (or Yang-Baxter) transformation, which may be used to gradually change the square lattice into any of these isoradial lattices, while preserving certain features of the model. It was previously proved that throughout this transformation, the large scale geometry of the model is distorted by at most a limited amount. In the present work we argue that the distortion becomes insignificant as the scale increases. This hinges on the interplay between the inhomogeneity of isoradial models and their embeddings, which compensate each other at large scales.
As a byproduct, we obtain the asymptotic rotational invariance also for models related to FK-percolation, such as the Potts and six-vertex ones. Moreover, the approach described here is fairly generic and may be adapted to other systems which possess a Yang-Baxter transformation. Based on joint work with Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun and Mendes Oulamara.
Our result is obtained via a universality theorem for FK-percolation on certain isoradial lattices. This in turn is proved via the star-triangle (or Yang-Baxter) transformation, which may be used to gradually change the square lattice into any of these isoradial lattices, while preserving certain features of the model. It was previously proved that throughout this transformation, the large scale geometry of the model is distorted by at most a limited amount. In the present work we argue that the distortion becomes insignificant as the scale increases. This hinges on the interplay between the inhomogeneity of isoradial models and their embeddings, which compensate each other at large scales.
As a byproduct, we obtain the asymptotic rotational invariance also for models related to FK-percolation, such as the Potts and six-vertex ones. Moreover, the approach described here is fairly generic and may be adapted to other systems which possess a Yang-Baxter transformation. Based on joint work with Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun and Mendes Oulamara.
Brownian half-plane excursions, CLE_4 and critical Liouville quantum gravity
Ellen Powell (Durham University)
8
I will discuss a coupling between a Brownian excursion in the upper half plane and an exploration of nested CLE_4 loops in the unit disk. In this coupling, the CLE_4 is drawn on top of an independent “critical Liouville quantum gravity surface” known as a quantum disk. It turns out that there is a correspondence between loops in the CLE and (sub) half-planar excursions above heights in the Brownian excursion, where the “width” of the sub-excursion corresponds to the “quantum boundary length” of the loop and the height encodes a certain “quantum distance” from the boundary.
This is based on a forthcoming joint work with Juhan Aru, Nina Holden and Xin Sun, and describes the analogue of Duplantier-Miller-Sheffield’s “mating-of-trees correspondence” in the critical regime.
This is based on a forthcoming joint work with Juhan Aru, Nina Holden and Xin Sun, and describes the analogue of Duplantier-Miller-Sheffield’s “mating-of-trees correspondence” in the critical regime.
Q&A for Invited Session 24
0
This talk does not have an abstract.
Session Chair
Nina Holden (Swiss Federal Institute of Technology Zürich)