We derive exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks. Our approach captures key features of extreme events in molecular motor motion along linear filaments. For one-dimensional biased random walks, we derive exact results which tighten bounds for entropy production extrema obtained with martingale theory and reveal a symmetry between the distribution of the maxima and minima of entropy production. Furthermore, we show that the relaxation spectrum of the full generating function, and hence of any moment, of the finite-time extrema distributions can be written in terms of the Marcenko-Pastur distribution of random-matrix theory. Using this result, we obtain efficient estimates for the extreme-value statistics of stochastic transport processes from the eigenvalue distributions of suitable Wishart and Laguerre random matrices. We confirm our results with numerical simulations of stochastic models of molecular motors.

中文翻译：

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随机传输过程的极值统计

我们推导出空间位移的极值（最大值和最小值）和有偏随机游走的波动熵流的有限时间统计的精确表达式。我们的方法捕捉分子运动沿线性细丝运动的极端事件的关键特征。对于一维有偏随机游走，我们推导出精确的结果，这些结果收紧了用鞅理论获得的熵产生极值的界限，并揭示了熵产生的最大值和最小值分布之间的对称性。此外，我们表明，有限时间极值分布的完整生成函数的弛豫谱以及任何时刻的弛豫谱都可以根据随机矩阵理论的 Marcenko-Pastur 分布来写。使用这个结果，我们从合适的 Wishart 和 Laguerre 随机矩阵的特征值分布中获得随机传输过程的极值统计的有效估计。我们通过分子马达随机模型的数值模拟证实了我们的结果。