We prove that the free energy of the log-gamma polymer between lattice points (1, 1) and (M, N) converges to the GUE Tracy–Widom distribution in the \(M^{1/3}\) scaling, provided that N/M remains bounded away from zero and infinity. We prove this result for the model with inverse gamma weights of any shape parameter \(\theta >0\) and furthermore establish a moderate deviation estimate for the upper tail of the free energy in this case. Finally, we consider a non i.i.d. setting where the weights on finitely many rows and columns have different parameters, and we show that when these parameters are critically scaled the limiting free energy fluctuations are governed by a generalization of the Baik–Ben Arous–Péché distribution from spiked random matrices with two sets of parameters.

我们证明格点 (1, 1) 和 ( M ,  N )之间的 log-gamma 聚合物的自由能在\(M^{1/3}\)缩放中收敛到 GUE Tracy-Widom 分布，前提是该ñ /中号保持从零至无穷大界远离。我们证明了具有任何形状参数\(\theta >0\) 的反伽马权重的模型的这个结果在这种情况下，进一步为自由能的上尾建立一个中等偏差估计。最后，我们考虑了一个非 iid 设置，其中有限多行和列上的权重具有不同的参数，并且我们表明，当这些参数被严格缩放时，限制自由能波动受 Baik-Ben Arous-Péché 分布的泛化控制来自具有两组参数的尖峰随机矩阵。