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We consider vector fixed point (FP) equations in large dimensional spacesinvolving random variables, and study their realization-wise solutions. We havean underlying directed random graph, that defines the connections betweenvarious components of the FP equations. Existence of an edge between nodes i, jimplies the i th FP equation depends on the j th component. We consider aspecial case where any component of the FP equation depends upon an appropriateaggregate of that of the random neighbor components. We obtain finitedimensional limit FP equations (in a much smaller dimensional space), whosesolutions approximate the solution of the random FP equations for almost allrealizations, in the asymptotic limit (number of components increase). Ourtechniques are different from the traditional mean-field methods, which dealwith stochastic FP equations in the space of distributions to describe thestationary distributions of the systems. In contrast our focus is onrealization-wise FP solutions. We apply the results to study systemic risk in alarge financial heterogeneous network with many small institutions and one biginstitution, and demonstrate some interesting phenomenon.

Random Fixed Points, Limits and Systemic risk