We study a translational invariant free fermions model in imaginary time, with nearest neighbor and next-nearest neighbor hopping terms, for a class of inhomogeneous boundary conditions. This model is known to give rise to limit shapes and arctic curves, in the absence of the next-nearest neighbor perturbation. The perturbation considered turns out to not be always positive, that is, the corresponding statistical mechanical model does not always have positive Boltzmann weights. We investigate how the density profile is affected by this nonpositive perturbation. We find that in some regions, the effects of the negative signs are suppressed, and renormalize to zero. However, depending on boundary conditions, new ``crazy regions'' emerge, in which minus signs proliferate, and the density of fermions is not in $[0,1]$ anymore. We provide a simple intuition for such behavior, and compute exactly the density profile both on the lattice and in the scaling limit.

中文翻译：

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非概率费米子极限形状

对于一类非齐次边界条件，我们研究了具有最近邻和次近邻跳跃项的虚时间平移不变自由费米子模型。众所周知，在没有下一个最近邻扰动的情况下，该模型会产生极限形状和北极曲线。所考虑的扰动结果并不总是正的，也就是说，相应的统计力学模型并不总是具有正的 Boltzmann 权重。我们研究了这种非正扰动如何影响密度剖面。我们发现在某些区域，负号的影响被抑制，并重新归一化为零。然而，根据边界条件，新的“疯狂区域”出现，其中负号激增，费米子的密度不再在 $[0,1]$ 中。