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Register a new userErgodicity of some dynamics of DNA sequences
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We define interacting particle systems on configurations of the integer lattice (with values in some finite alphabet) by the superimposition of two dynamics: a substitution process with finite range rates, and a circular permutation mechanism(called cut-and-paste) with possibly unbounded range. The model is motivated by the dynamics of DNA sequences: we consider an ergodic model for substitutions, the RN+YpR model ([BGP08]), with three particular cases, the models JC+cpg,T92+cpg, and RNc+YpR. We investigate whether they remain ergodic with the additional cut-and-paste mechanism, which models insertions and deletions of nucleotides. Using either duality or attractiveness techniques, we provide various sets of sufficient conditions, concerning only the substitution rates, for ergodicity of the superimposed process. They imply ergodicity of the models JC+cpg, T92+cpg as well as the attractive RNc+YpR, all with an additional cut-and-paste mechanism.
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We study the long time behavior of an underdamped mean-field Langevin (MFL) equation, and provide a general convergence as well as an exponential convergence rate result under different conditions. The results on the MFL equation can be applied to study the convergence of the Hamiltonian gradient descent algorithm for the overparametrized optimization. We then provide a numerical example of the algorithm to train a generative adversarial networks (GAN).
Max-stable processes are central models for spatial extremes. In this paper, we focus on some space-time max-stable models introduced in Embrechts et al. (2016). The processes considered induce discrete-time Markov chains taking values in the space of continuous functions from the unit sphere of $mathbb{R}^3$ to $(0, infty)$. We show that these Markov chains are geometrically ergodic. An interesting feature lies in the fact that the state space is not locally compact, making the classical methodology inapplicable. Instead, we use the fact that the state space is Polish and apply results presented in Hairer (2010).
We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such as the functions $t^{3/2}, tlog t$ and $e^{sqrt{log t}}$. We show that if all non-trivial linear combinations of the functions $a_1,...,a_k$ stay logarithmically away from rational polynomials, then the $L^2$-limit of the ergodic averages $frac{1}{N} sum_{n=1}^{N}f_1(T^{lfloor{a_1(n)}rfloor}x)cdots f_k(T^{lfloor{a_k(n)}rfloor}x)$ exists and is equal to the product of the integrals of the functions $f_1,...,f_k$ in ergodic systems, which establishes a conjecture of Frantzikinakis. Under some more general conditions on the functions $a_1,...,a_k$, we also find characteristic factors for convergence of the above averages and deduce a convergence result for weak-mixing systems.
We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure preserving $mathbb{Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and a null sequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on $mathbb{Z}^{d}$-systems.
The Northeast Model is a spin system on the two-dimensional integer lattice that evolves according to the following rule: Whenever a sites southerly and westerly nearest neighbors have spin $1$, it may reset its own spin by tossing a $p$-coin; at all other times, its spin remains frozen. It is proved that the northeast model has a phase transition at $p_{c}=1-beta_{c}$, where $beta_{c}$ is the critical parameter for oriented percolation. For $p<p_{c}$, the trivial measure $delta_{0}$ that puts mass one on the configuration with all spins set at $0$ is the unique ergodic, translation invariant, stationary measure. For $pgeq p_{c}$, the product Bernoulli-$p$ measure on configuration space is the unique nontrivial, ergodic, translation invariant, stationary measure for the system, and it is mixing. For $p>2/3$ it is shown that there is exponential decay of correlations.
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