Jack变形随机Young图在不同温度下的全局渐近性的普遍性
Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures
AI总结：
本文研究了Jack变形随机Young图在不同温度下的全局渐近性，揭示了其极限形状和波动的普遍性质，证明了在高温和低温下极限形状为一侧无限阶梯形。
摘要：
本文建立了描述两种不同的β-集合的离散版本在高温、低温和固定温度极限下的全局渐近性的通用公式。我们的结果肯定地回答了第二作者和Śniady提出的问题。我们首先引入了一种特殊的Jack测度，称为"Jack-Thoma测度"，并证明了在上述三种极限情况下大数定律和中心极限定理。在每种情况下，我们为极限形状的多项式观测值和围绕极限形状的高斯波动提供了显式公式。这些公式具有令人惊讶的正性性质，并以加权格路之和的形式表达。其次，我们证明了这些公式是通用的：它们也描述了由Jack特征导出的固定大小随机Young图模型的极限形状和高斯波动。最后，与连续β-集合形成鲜明对比的是，我们展示了我们随机Young图在高温和低温下的极限形状是一侧无限阶梯形状。对于Jack-Plancherel测度，我们通过将其局部极小值与Bessel函数的零点相关联，明确描述了这种形状。
英文摘要：
This paper establishes universal formulas describing the global asymptotics of two distinct discrete versions of $β$-ensembles in the high, low and fixed temperature regimes. Our results affirmatively answer a question posed by the second author and Śniady. We first introduce a special class of Jack measures on Young diagrams of arbitrary size, called the ``Jack--Thoma measures'', and prove the LLN and CLT in the three aforementioned limit regimes. In each case, we provide explicit formulas for polynomial observables of the limit shape and Gaussian fluctuations around the limit shape. These formulas have surprising positivity properties and are expressed as sums of weighted lattice paths. Second, we show that the previous formulas are universal: they also describe the limit shape and Gaussian fluctuations for the model of random Young diagrams of a fixed size derived from Jack characters with the approximate factorization property. Finally, in stark contrast with continuous $β$-ensembles, we show that the limit shapes at high and low temperatures of our random Young diagrams are one-sided infinite staircase shapes. For the Jack--Plancherel measure, we describe this shape explicitly by relating its local minima with the zeroes of Bessel functions.
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