标题： 随机图上相互作用粒子系统的相变 显示英文标题

标题： Phase transitions for interacting particle systems on random graphs

摘要： 本文中，我们研究了在随机图上的弱相互作用扩散过程。 我们的主要关注点在于平均场极限的性质，特别是稳态解的非唯一性。 通过将经典的分支分析扩展到包含多色交互势和随机图结构，我们明确地识别出分支点，并将其与图论积分算子的特征值联系起来。 此外，我们将所得到的McKean-Vlasov偏微分方程表征为相对于适当度量的梯度流。 我们结合这些理论结果与线性化McKean-Vlasov算子的谱分析以及广泛的数值模拟，以深入了解稳态解的稳定性和长期行为。 此外，我们提供了有力的证据表明，（减去的）相互作用粒子系统的交互能作为一个自然的序参量。 特别是，在过渡点之外且对于多色交互时，我们观察到了一个能量级联，这与系统的动态亚稳态密切相关。 显示英文摘要

摘要： In this paper, we study weakly interacting diffusion processes on random graphs. Our main focus is on the properties of the mean-field limit and, in particular, on the nonuniqueness of stationary states. By extending classical bifurcation analysis to include multichromatic interaction potentials and random graph structures, we explicitly identify bifurcation points and relate them to the eigenvalues of the graphon integral operator. Furthermore, we characterize the resulting McKean-Vlasov PDE as a gradient flow with respect to a suitable metric. We combine these theoretical results with the spectral analysis of the linearized McKean-Vlasov operator and extensive numerical simulations to gain insight into the stability and long-term behaviour of stationary solutions. In addition, we provide strong evidence that (minus) the interaction energy of the interacting particle system serves as a natural order parameter. In particular, beyond the transition point and for multichromatic interactions, we observe an energy cascade that is strongly linked to the dynamical metastability of the system.