随机平面图中的新相变与RNA型匹配
New phase transition in random planar diagrams and RNA-type matching
摘要：
我们研究了平面匹配问题，定义了一个对称随机矩阵与独立同分布的项目，取值0和1。我们表明，一个完美的平面匹配结构的存在是可能的，只有在一定的临界密度，$p_{c}$，允许接触（即“1”）。使用代克路径和平面接触结构的矩阵模型的问题的制定，我们提供了一个分析估计的值的过渡点，$p_{c}$，在热力学极限。该估计接近临界值，$p_{c} \approx 0.379$，在基于精确动态规划算法的数值模拟中获得。我们描述了相应的临界行为的模型和讨论的关系的完美-不完美匹配过渡到已知的熔融玻璃化转变的背景下随机RNA二级结构的形成。特别是，我们提供了强有力的证据支持的猜想，熔融玻璃化转变在T=0发生在$p_{c}$。
英文摘要：
We study the planar matching problem, defined by a symmetric random matrix with independent identically distributed entries, taking values 0 and 1. We show that the existence of a perfect planar matching structure is possible only above a certain critical density, $p_{c}$, of allowed contacts (i.e. of '1'). Using a formulation of the problem in terms of Dyck paths and a matrix model of planar contact structures, we provide an analytical estimation for the value of the transition point, $p_{c}$, in the thermodynamic limit. This estimation is close to the critical value, $p_{c} \approx 0.379$, obtained in numerical simulations based on an exact dynamical programming algorithm. We characterize the corresponding critical behavior of the model and discuss the relation of the perfect-imperfect matching transition to the known molten-glass transition in the context of random RNA secondary structure's formation. In particular, we provide strong evidence supporting the conjecture that the molten-glass transition at T=0 occurs at $p_{c}$.
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