We introduce a planted vertex cover problem on regular random graphs and study it by the cavity method. The equilibrium ordering phase transition of this binary-spin two-body interaction system is discontinuous in nature distinct from the continuous one of conventional Ising-like models, and it is dynamically blocked by an extensive free energy barrier. We discover that the disordered symmetric phase of this system may be locally stable with respect to the ordered phase at all inverse temperatures except for a unique eureka point $\beta_b$ at which it is only marginally stable. The eureka point $\beta_b$ serves as a backdoor to access the hidden ground state with vanishing free energy barrier. It exists in an infinite series of planted random graph ensembles and we determine their structural parameters analytically. The revealed new type of free energy landscape may also exist in other planted random-graph optimization problems at the interface of statistical physics and statistical inference.

翻译：我们引入了一个在规则随机图上的植入顶点覆盖问题，并通过空穴法进行研究。这一二元自旋双体相互作用系统的平衡有序相变在本质上是不连续的，不同于传统伊辛类模型的连续相变，并且被广泛自由能势垒动态阻碍。我们发现，该系统的无序对称相在所有逆温度下相对于有序相可能局部稳定，除了一个独特的诱拐点$\beta_b$，在该点它仅处于边缘稳定状态。诱拐点$\beta_b$作为一个后门，以零自由能势垒进入隐藏基态。它存在于无限系列的植入随机图系综中，我们解析确定了其结构参数。所揭示的这一新型自由能景观也可能存在于统计物理与统计推断交叉领域的其他植入随机图优化问题中。