随机正则图中的相互作用热场偶和临界行为
Interacting Thermofield Doubles and Critical Behavior in Random Regular Graphs
摘要：
本文用数值方法讨论了指数随机图中的非微扰效应，指数随机图是矩阵模型中本征值瞬子的模拟。阐明了具有4圈化学势和度保持约束的指数随机图的相结构。研究了无规正则图系综在化学势临界值处的一级相变，发现4-圈相变为二分相，并形成固定数目的二分团簇。我们考虑了组合量子引力中的类似相变，并基于补充了硬核约束的RRG的Ollivier图曲率，证明了相变的阶数和出现的相的结构依赖于RRG中的顶点度d.当d = 3时，在二分相出现二分封闭带，而当d > 3时，在一级相变出现孤立或弱相互作用的超立方体系综，并伴有明显的滞后。如果附加连通性条件，则二分相被标识为弱耦合超立方体的闭链。由于孤立超立方体的基态是热场偶（TFD），我们认为对偶全息图包含多边界虫洞。将RRG作为相互作用多体系统的Hilbert空间模型，讨论了在相变时Hilbert空间的碎裂模式。我们还简要地评论了一个可能的关系，发现相变的问题的全息解释的部分解除禁闭过渡的规范理论。
英文摘要：
We discuss numerically the non-perturbative effects in exponential random graphs which are analogue of eigenvalue instantons in matrix models. The phase structure of exponential random graphs with chemical potential for 4-cycles and degree preserving constraint is clarified. The first order phase transition at critical value of chemical potential for 4-cycles into bipartite phase with a formation of fixed number of bipartite clusters is found for ensemble of random regular graphs (RRG). We consider the similar phase transition in combinatorial quantum gravity based of the Ollivier graph curvature for RRG supplemented with hard-core constraint and show that a order of a phase transition and the structure of emerging phase depend on a vertex degree d in RRG. For d = 3 the bipartite closed ribbon emerges at bipartite phase while for d > 3 the ensemble of isolated or weakly interacting hypercubes supplemented with the bipartite closed ribbon gets emerged at the first order phase transition with a clear-cut hysteresis. If the additional connectedness condition is imposed the bipartite phase gets identified as the closed chain of weakly coupled hypercubes. Since the ground state of isolated hypercube is the thermofield double (TFD) we suggest that the dual holographic picture involves multiboundary wormholes. Treating RRG as a model of a Hilbert space for a interacting many-body system we discuss the patterns of the Hilbert space fragmentation at the phase transition. We also briefly comment on a possible relation of the found phase transition to the problem of holographic interpretation of a partial deconfinement transition in the gauge theories.
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