二聚体、滤波器与q变形实数
Dimers, filters, and $q$-deformed real numbers
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中文总结 AI 辅助
研究将正实数x与含活动参数q>0的非齐次二聚体模型关联，定义函数q↦[[x]]_q，与代数q变形[x]_q相关。用偏序集滤波器等效模型阐述技术细节，有理数时[[x]]_q = q [x]_q ，猜测一般情况下方程两边在有定义处一致。
中文摘要 AI 辅助
本文将每个正实数x与一个具有活动参数q>0的非齐次二聚体模型相关联，并用它来定义一个正实值函数q↦[[x]]_q，该函数与Morier-Genoud和Ovsienko引入的代数q变形[x]_q相关（实际上是受其启发）。当二聚体模型被涉及偏序集中滤波器的等效模型取代时，技术细节最为清晰。当x为有理数时，[[x]]_q = q [x]_q，并且更一般地，方程两边在都有定义的地方似乎是一致的。
英文摘要
This article associates to each positive real number $x$ a dimer model on a snake graph with activity parameter $q>0$ whose structure is determined by the continued fraction expansion of $x$. When $x$ is rational, the model is finite and gives rise to a probability measure $μ_{x,q}$ on perfect matchings. For irrational $x$, the model is infinite, and $μ_{x,q}$ is defined as a limit over rational approximations to $x$; the main technical result of the paper shows that this limit is well defined and independent of the choice of rational approximants. $[[x]]_q$ denotes the odds that a $μ_{x,q}$-random perfect matching includes a distinguished edge. When $x$ is rational, $[[x]]_q = q\:[x]_q$, where $[x]_q$ is the algebraic $q$-deformation introduced by Morier-Genoud and Ovsienko. This agreement, together with evidence from the irrational case, suggests a close connection between the probabilistic and algebraic constructions.
