Related papers: Mixing, wine, and serendipity
Three topics in dynamical systems are discussed. In the first two sections we solve some open problems concerning, respectively, Furstenberg entropy of stationary dynamical systems, and uniformly rigid actions admitting a weakly mixing…
We derive sufficient conditions for the mixing of all orders of interacting transformations of a spatial Poisson point process, under a zero-type condition in probability and a generalized adaptedness condition. This extends a classical…
By benefit of Pesin's method to prove ergodicity with respect to Lebesgue measure for ordinary dynamical systems, we conclude ergodicity (resp. term-ergodicity) for some action semigroups with respect to volume measure (resp. quasi…
The aim of this work is to put forward a statistical mechanics theory of social interaction, generalizing econometric discrete choice models. After showing the formal equivalence linking econometric multinomial logit models to equilibrium…
In this paper we develop the theory of {\it polymorphisms} of measure spaces, which is a generalization of the theory of measure-preserving transformations; we describe the main notions and discuss relations to the theory of Markov…
The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for sigma-finite invariant measures (Corollary 1). For…
In this monograph we lay the foundation for a theory of coarse groups and coarse actions. Coarse groups are group objects in the category of coarse spaces, and can be thought of as sets with operations that satisfy the group axioms "up to…
Evolutionary game theory is a common framework to study the evolution of cooperation, where it is usually assumed that the same game is played in all interactions. Here, we investigate a model where the game that is played by two…
After a brief historical review of ergodicity and mixing in dynamics, particularly in quantum dynamics, we introduce definitions of quantum ergodicity and mixing using the structure of the system's energy levels and spacings. Our…
In its general definition, the law of mass action posits that, in systems where multiple elements move randomly, the rate by which they physically interact is proportional to the product of their densities. This law predicts the rate of…
We consider a random walk $S_k$ with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum $\sum_{k=1}^N f(S_k)$ with H\"older continuous test functions $f$, including the…
Apportionment is the act of distributing the seats of a legislature among political parties (or states) in proportion to their vote shares (or populations). A famous impossibility by Balinski and Young (2001) shows that no apportionment…
Let $l\in \mathbb{N}_{\geq 1}$ and $\alpha : \mathbb{Z}^l\rightarrow \text{Aut}(\mathscr{N})$ be an action of $\mathbb{Z}^l$ by automorphisms on a compact nilmanifold $\mathscr{N}$. We assume the action of every $\alpha(z)$ is ergodic for…
We introduce a new class of sparse sequences that are ergodic and pointwise universally $L^2$-good for ergodic averages. That is, sequences along which the ergodic averages converge almost surely to the projection to invariant functions.…
How many shuffles are needed to mix up a deck of cards? This question may be answered in the language of a random walk on the symmetric group, $S_{52}$. This generalises neatly to the study of random walks on finite groups, themselves a…
Time-asymmetric behavior as embodied in the second law of thermodynamics is observed in {\it individual macroscopic} systems. It can be understood as arising naturally from time-symmetric microscopic laws when account is taken of a) the…
The paper deals with the problem of long-time asymptotic behaviour of solutions for classes of ODEs and PDEs, perturbed by stationary noises. The latter are not assumed to be $\delta$-correlated in time, so that the evolution in question is…
Measures of economic mobility represent aggregate values for how individual wealth changes over time. As such, these measures may not describe the feasibility of a typical individual to change their wealth. To address this limitation, we…
In this article we will see some properties that guarantee that a product of an ergodic non-singular action and a probability preserving ergodic action is also an ergodic action. We will start by proving 'The multiplier theorem' for locally…
We consider the averaging process on an infinite connected graph with bounded degree and independent, identically distributed starting values or initial opinions. Assuming that the law of the initial opinion of a vertex has a finite second…