标题： 边界驱动零点过程在图上的当前波动：微观与宏观方法以及非可逆电阻类似网络的理论 显示英文标题

标题： Current fluctuations for the boundary-driven zero-range process on graphs: microscopic versus macroscopic approach and a theory of non-reversible resistor-like networks

摘要： 我们计算了在大时间极限下，有限图的边上的电流的联合大偏差率泛函，对于边界驱动的零范围动力学。 这推广了之前用不同方法获得的一维结果\cite{BDGJL1,HRS}；我们的替代技术阐明了各种联系和互补视角。 特别是，我们在这里使用变分方法，通过从一个二级2.5大偏差率泛函收缩得到率泛函。 我们进行了精确的最小化，最终得到的率泛函是一个涉及每条边成本函数叠加的变分问题。 不同边的贡献不是独立的，因为它们由图节点上的势函数值相关联。 图上的率泛函是宏观波动理论预测的连续率泛函的微观版本\cite{MFT}，并且我们确实展示了在尺度极限下的收敛性。 如果我们通过一个割集将图分成两个连通区域，并且只关心通过割集的电流，我们会发现结果与仅由一个有效边组成的有效系统的结果相同（如宏观层次所发生的情况，也预期其他模型也是如此\cite{Cap}）。 这个有效边的特性与图的“容量”有关，可以通过使用类似于电路网络中的基本变换进行约简得到；具体来说，我们处理并联、串联以及$N$-星形配置的组件（约简为有效的完全$N$-图）。 我们的约简过程直接与迹过程的约简相关\cite{L}，并且由于动力学通常不可逆，它也与非可逆电路网络理论在\cite{B}中密切相关。 显示英文摘要

摘要： We compute the joint large deviation rate functional in the limit of large time for the current flowing through the edges of a finite graph for a boundary-driven zero-range dynamics. This generalizes one-dimensional results previously obtained with different approaches \cite{BDGJL1,HRS}; our alternative techniques illuminate various connections and complementary perspectives. In particular, we here use a variational approach to derive the rate functional by contraction from a level 2.5 large deviation rate functional. We perform an exact minimization and finally obtain the rate functional as a variational problem involving a superposition of cost functions for each edge. The contributions from different edges are not independent since they are related by the values of a potential function on the nodes of the graph. The rate functional on the graph is a microscopic version of the continuous rate functional predicted by the macroscopic fluctuation theory \cite{MFT}, and we indeed show a convergence in the scaling limit. If we split the graph into two connected regions by a cutset and are interested just in the current flowing through the cutset, we find that the result is the same as that of an effective system composed of only one effective edge (as happens at macroscopic level and is expected also for other models \cite{Cap}). The characteristics of this effective edge are related to the ``capacities'' of the graph and can be obtained by a reduction using elementary transformations as in electrical networks; specifically, we treat components in parallel, in series, and in $N$-star configurations (reduced to effective complete $N$-graphs). Our reduction procedure is directly related to the reduction to the trace process \cite{L} and, since the dynamics is in general not reversible, it is also closely connected to the theory of non-reversible electrical networks in \cite{B}.