对数REM的统计力学：高斯自由场产生的$1/f$噪声的对偶、冻结和极值统计
Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of $1/f$ Noises generated by Gaussian Free Fields
摘要：
本文计算了一类具有热力学关联随机势的一维随机能量模型（REM）在玻璃化温度以上和玻璃化温度下的配分函数分布。随机势序列表示通过沿着各种平面曲线对二维高斯自由场（2dGFF）进行采样而生成的1/f噪声的各种版本。我们的方法扩展了最近的分析Fyodorov Bouchaud从圆形的情况下，一个区间，是基于分析的连续的Selberg积分。特别地，我们揭示了一个{ \it对偶关系}所满足的适当的生成函数的自由能累积量在高温阶段。它加强了冻结的情况下，生成函数的假设，从中我们得到的分布极值的$[0,1]$间隔上的2dGFF。我们提供了数值检查的圆形和区间的情况下，并讨论了普遍性和各种扩展。指出了与刘维尔量子引力中一段长度分布的相关性。
英文摘要：
We compute the distribution of the partition functions for a class of one-dimensional Random Energy Models (REM) with logarithmically correlated random potential, above and at the glass transition temperature. The random potential sequences represent various versions of the 1/f noise generated by sampling the two-dimensional Gaussian Free Field (2dGFF) along various planar curves. Our method extends the recent analysis of Fyodorov Bouchaud from the circular case to an interval and is based on an analytical continuation of the Selberg integral. In particular, we unveil a {\it duality relation} satisfied by the suitable generating function of free energy cumulants in the high-temperature phase. It reinforces the freezing scenario hypothesis for that generating function, from which we derive the distribution of extrema for the 2dGFF on the $[0,1]$ interval. We provide numerical checks of the circular and the interval case and discuss universality and various extensions. Relevance to the distribution of length of a segment in Liouville quantum gravity is noted.
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