Veranstaltungen

In systems with positive feedback, microscopic changes cause macroscopic effects, cascading up to length scales determined by the feedback range. Long-range feedback, in particular, generate cascades that can propagate at all distances resulting in abrupt transitions with critical scaling. These intriguing transitions, often called hybrid or mixed-order [1, 2], have been reported somewhere theoretically, somewhere numerically on both random and spatial graphs, typically in ad hoc models. In this talk, we will show that their critical phenomena can be cast in a coherent statistical mechanics framework, predicting two universality classes of mixed-order transitions by long-range cascades defined by the parity invariance of the underlying process [3]. We will provide finite-size scaling arguments based on hyperscaling above upper critical dimensions [4], predicting critical exponents having both mean-field and $d$-dimensional features –hence, their term "chimeric''– for any $d\geq2$ and show how parity invariance influences the geometry and lifetime of avalanches. We will demonstrate the validity of such framework in several synthetic and experimentally-driven cascade models, with a particular emphasis on interdependent processes [5, 6], given their recent observation in laboratory experiments [7]. [1] R. M. D’Souza, J. Gómez-Gardenes, J. Nagler, and A. Arena, Advances in Physics, 68(3):123–223, 2019. [2] C. Kuehn and C. Bick, Science advances, 7(16):eabe3824, 2021. [3] I. B., B. Gross, J.Kertész, S. Havlin, arXiv preprint arXiv:2401.09313, 2024. [4] B. Berche, T. Ellis, Y. Holovatch, and R. Kenna, SciPost Physics Lecture Notes, 060, 2022. [5] S. V. Buldyrev et al., Nature 464.7291, 1025-1028, 2010. [6] M. M. Danziger, I. B., S. Boccaletti, S. Havlin, Nature Physics, 15(2), 178-185, 2019. [7] I.B. et al.,Nature Physics 19, 1163–1170 (2023).

Abstract:This work offers a derivation of the Vlasov equation from fermionic many-body Schrödinger systems, utilizing the Husimi measure as a connecting tool between classical mechanics and quantum mechanics. We start with an intuitive overview of the Vlasov equation, followed by a concise investigation of the many-body Schrödinger equation. The core of our discussion is about the usage of the Husimi measures to bridge these two equations. Participants will be introduced to the underlying formalism and techniques for the derivation process.

In first-price and all-pay auctions under the standard symmetric independent private-values model, we show that the unique Bayesian Coarse Correlated Equilibrium with symmetric, differentiable and strictly increasing bidding strategies is the unique strict Bayesian Nash Equilibrium. Interestingly, this result does not require assumptions on the prior distribution. The proof is based on a dual bound of the infinite-dimensional linear program. Numerical experiments without restrictions on bidding strategies show that for first-price auctions and discretisations up to 21 of the type and bid space, increasing discretisation sizes actually increase the concentration of Bayesian Coarse Correlated Equilibrium over the Bayesian Nash Equilibrium, so long as the prior c.d.f. is concave. Such a concentration is also observed for all-pay auctions, independent of the prior distribution. Overall, our results imply that the equilibria of these important class of auctions are indeed learnable.

In this talk, I am going to outline the recent history of the theory of early-warning signs for differential equations with a focus on stochastic dynamics near bifurcations. A particular emphasis will be put on the intertwining of the development of this theory with the simultaneous surge of interest in understanding applied geophysical systems and their tipping points.