Holehouse James
The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA.
Entropy (Basel). 2023 Jun 29;25(7):996. doi: 10.3390/e25070996.
Understanding how systems relax to equilibrium is a core theme of statistical physics, especially in economics, where systems are known to be subject to extrinsic noise not included in simple agent-based models. In models of binary choice-ones not much more complicated than Kirman's model of ant recruitment-such relaxation dynamics become difficult to determine analytically and require solving a three-term recurrence relation in the eigendecomposition of the stochastic process. In this paper, we derive a concise closed-form solution to this linear three-term recurrence relation. Its solution has traditionally relied on cumbersome continued fractions, and we instead employ a linear algebraic approach that leverages the properties of lower-triangular and tridiagonal matrices to express the terms in the recurrence relation using a finite set of orthogonal polynomials. We pay special attention to the power series coefficients of Heun functions, which are also important in fields such as quantum mechanics and general relativity, as well as the binary choice models studied here. We then apply the solution to find equations describing the relaxation to steady-state behavior in social choice models through eigendecomposition. This application showcases the potential of our solution as an off-the-shelf solution to the recurrence that has not previously been reported, allowing for the easy identification of the eigenspectra of one-dimensional, one-step, continuous-time Markov processes.
理解系统如何弛豫到平衡态是统计物理学的一个核心主题，在经济学中尤其如此，因为在经济学中，已知系统会受到简单基于主体的模型中未包含的外部噪声的影响。在二元选择模型中——这些模型并不比基尔曼的蚂蚁招募模型复杂多少——这种弛豫动力学很难通过解析方法确定，并且需要求解随机过程特征分解中的一个三项递推关系。在本文中，我们推导出了这个线性三项递推关系的一个简洁的闭式解。其解传统上依赖于繁琐的连分数，而我们转而采用一种线性代数方法，该方法利用下三角矩阵和三对角矩阵的性质，用有限集的正交多项式来表示递推关系中的项。我们特别关注合流超几何函数的幂级数系数，其在量子力学和广义相对论等领域以及本文研究的二元选择模型中也很重要。然后，我们应用该解，通过特征分解来找到描述社会选择模型中弛豫到稳态行为的方程。这一应用展示了我们的解作为该递推关系的一个现成解的潜力，此前尚未有过报道，它使得一维、一步、连续时间马尔可夫过程的本征谱易于识别。
