arXiv 2203.024432022-03-04math.PRmath.CO

循环生灭链中的循环积与最优陷阱

Cyclic products and optimal traps in cyclic birth and death chains

Mark Holmes, Alexander E. Holroyd, Alejandro Ramírez

备注：: 20 pages

摘要：

生灭链是一个离散时间马尔可夫链的整数，其转移概率$p_{i,j}$是非零的当且仅当$|i-j|=1$。我们考虑生灭链，其出生概率$p_{i,i+1}$形成周期序列，使得对于某些$m$和$p_0,\ldots,p_{m-1}$，$p_{i,i+1}=p_{i \mod m}$.这种链的轨迹$(X_n)_{n=0,1,\ldots}$满足一个强大数定律和一个中心极限定理。研究了概率$p_0,\ldots,p_{m-1}$重排对速度$v=\lim_{n\to\infty} X_n/n$的影响。$v$的符号不受重新排序的影响，但它的大小通常会受到影响。我们表明，勒贝格几乎每一个选择的$(p_0,\ldots,p_{m-1})$，正是$(m-1)!/2$不同的速度可以通过重新排序。我们提出了一个明确的猜想的顺序，最小化的速度，并证明它的所有$m\leq 7$。这个猜想是由一个纯粹的组合猜想，我们认为是独立的利益。

英文摘要：

A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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