We obtain an exact analytic expression for the average distribution, in the thermodynamic limit, of overlaps between two copies of the same random energy model (REM) at different temperatures. We quantify the non-self averaging effects and provide an exact approach to the computation of the fluctuations in the distribution of overlaps in the thermodynamic limit. We show that the overlap probabilities satisfy recurrence relations that generalise Ghirlanda–Guerra identities to two temperatures. We also analyse the two temperature REM using the replica method. The replica expressions for the overlap probabilities satisfy the same recurrence relations as the exact form. We show how a generalisation of Parisi’s replica symmetry breaking ansatz is consistent with our replica expressions. A crucial aspect to this generalisation is that we must allow for fluctuations in the replica block sizes even in the thermodynamic limit. This contrasts with the single temperature case where the extremal condition leads to a fixed block size in the thermodynamic limit. Finally, we analyse the fluctuations of the block sizes in our generalised Parisi ansatz and show that in general they may have a negative variance.

我们获得了在热力学极限下，相同随机能量模型（REM）的两个副本在不同温度下的重叠平均分布的精确解析表达式。我们量化了非自我平均效应，并提供了一种精确的方法来计算热力学极限中重叠分布的波动。我们表明，重叠概率满足将Ghirlanda-Guerra身份推广到两个温度的递归关系。我们还使用复制方法分析了两个温度REM。重叠概率的副本表达式满足与精确形式相同的递归关系。我们展示了Parisi的复制对称性打破ansatz的推广与我们的复制表达相一致的方式。这种概括的一个关键方面是，即使在热力学极限内，我们也必须允许复制块大小的波动。这与极端温度导致热力学极限中的固定块尺寸的单一温度情况相反。最后，我们分析了广义Parisi ansatz中块大小的波动，并表明它们总体上可能具有负方差。