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In this paper, the problem of distributed dynamic frequency allocation is considered for a canonical communication network, which spans several networks such as cognitive radio networks and digital subscriber lines (DSLs). A greedy asynchronous distributed interference avoidance (GADIA) algorithm for horizontal spectrum sharing has been proposed that achieves performance close to that of a centralized optimal algorithm. The convergence of the GADIA algorithm to a near-optimal frequency allocation strategy is proved and several asymptotic performance bounds have been established for various spatial configurations of the network nodes. Furthermore, the near-equilibrium dynamics of the GADIA algorithm has been studied using the Glauber dynamics, by identifying the problem with the antiferromagnetic inhomogeneous long-range Potts model. Using the near-equilibrium dynamics and methods from stochastic analysis, the robustness of the algorithm with respect to time variations in the activity of network nodes is studied. These analytic results along with simulation studies reveal that the performance is close to that of an optimum centralized frequency allocation algorithm. Further simulation studies confirm that our proposed algorithm outperforms the iterative water-filling algorithm in the low signal-to-interference-plus-noise ratio (SINR) regime, in terms of achieved sum rate, complexity, convergence rate, and robustness to time-varying node activities.