Perfect simulation and finitary coding for multicolor systems with interactions of infinite range

Finitary codings for gradient models and a new graphical representation for the six-vertex model

It is known that the Ising model on $\mathbb {Z}^d$ at a given temperature is a finitary factor of an i.i.d. process if and only if the temperature is at least the critical temperature. Below the critical temperature, the plus and minus states of the Ising model are distinct and differ from one another by a global flip of the spins. We show that it is only this global information which poses an obstruction for being finitary by showing that the gradient of the Ising model is a finitary factor of i.i.d. at all temperatures. As a consequence, we deduce a volume-order large deviation estimate for…

泊松可表示过程的有限编码与随机占优

Finitary codings and stochastic domination for Poisson representable processes

本文研究由整数有限子集按平移不变概率独立选取并取并得到的随机集合，证明其在所选集合大小分布具有指数矩时是IID过程的有限因子，并刻画了其被伯努利渗流随机占优的充要条件及相变现象。

Finitary codings for spatial mixing Markov random fields

It has been shown by van den Berg and Steif that the sub-critical and critical Ising model on $\mathbb{Z}^d$ is a finitary factor of an i.i.d. process (ffiid), whereas the super-critical model is not. In fact, they showed that the latter is a general phenomenon in that a phase transition presents an obstruction for being ffiid. The question remained whether this is the only such obstruction. We make progress on this, showing that certain spatial mixing conditions (notions of weak dependence on boundary conditions, not to be confused with other notions of mixing in ergodic theory) imply ffiid. …

有限码与随机场的高斯集中性

Finitary coding and Gaussian concentration for random fields

本文研究了通过有限码编码得到的随机场的高斯集中性不等式，探讨了集中性与编码结构的关系。主要结果表明，在有限二阶矩条件下，有限码保持高斯集中性。应用结果到Gibbs测度、Z^d上的马尔可夫随机场及一维过程，得出经典晶格模型中高斯集中性的充分必要条件。

An invariant of finitary codes with finite expected square root coding length

Let $p$ and $q$ be probability vectors with the same entropy $h$. Denote by $B(p)$ the Bernoulli shift indexed by $\Z$ with marginal distribution $p$. Suppose that $ϕ$ is a measure preserving homomorphism from $B(p)$ to $B(q)$. We prove that if the coding length of $ϕ$ has a finite 1/2 moment, then $σ_p^2=σ_q^2$, where $σ_p^2=\sum_i p_i(-\log p_i-h)^2$ is the {\dof informational variance} of $p$. In this result, which sharpens a theorem of Parry (1979), the 1/2 moment cannot be replaced by a lower moment. On the other hand, for any $θ<1$, we exhibit probability vectors $p$ and $q$ that are not…

One-dependent coloring by finitary factors

Holroyd and Liggett recently proved the existence of a stationary 1-dependent 4-coloring of the integers, the first stationary k-dependent q-coloring for any k and q. That proof specifies a consistent family of finite-dimensional distributions, but does not yield a probabilistic construction on the whole integer line. Here we prove that the process can be expressed as a finitary factor of an i.i.d. process. The factor is described explicitly, and its coding radius obeys power-law tail bounds.

通过随机占优与有限编码刻画可和性

Characterizations of amenability through stochastic domination and finitary codings

本文利用随机占优、不变单调耦合与有限因子，刻画图的不可和性，并将极低温 Ising 正态及高密度 Bernoulli 渗流无限簇过程联系起来。

Bernoulli过程之间有限因子映射的矩

Moments of finitary factor maps between Bernoulli processes

本文研究Zd过程间有限因子映射编码半径的矩，推广Harvey-Peres结果，证明d/2阶矩有限蕴含信息方差相等，并给出指数尾部下的Schmidt型定理。

Attractive regular stochastic chains: perfect simulation and phase transition

We prove that uniqueness of the stationary chain, or equivalently, of the $g$-measure, compatible with an attractive regular probability kernel is equivalent to either one of the following two assertions for this chain: (1) it is a finitary coding of an i.i.d. process with countable alphabet, (2) the concentration of measure holds at exponential rate. We show in particular that if a stationary chain is uniquely defined by a kernel that is continuous and attractive, then this chain can be sampled using a coupling-from-the-past algorithm. For the original Bramson-Kalikow model we further prove t…

等熵i.i.d.过程的矩最优有限性同构

Moment-optimal finitary isomorphism for i.i.d. processes of equal entropy

本文解决了等熵i.i.d.过程有限性同构的矩问题，显式构造了在任意两个等熵非周期Markov过程间对所有t∈(0,1/2)具有有限t-矩的同构，并证明其尾部在多对数因子内最优。

Mallows Permutations and Finite Dependence

We use the Mallows permutation model to construct a new family of stationary finitely dependent proper colorings of the integers. We prove that these colorings can be expressed as finitary factors of i.i.d. processes with finite mean coding radii. They are the first colorings known to have these properties. Moreover, we prove that the coding radii have exponential tails, and that the colorings can also be expressed as functions of countable-state Markov chains. We deduce analogous existence statements concerning shifts of finite type and higher-dimensional colorings.