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Congested hot spots and node failures severely degrade the performance of wireless mesh networks. However, conventional routing schemes are inefficient in mitigation of the problems. Considering analogy to physics, we propose a novel distributed potential-field-based routing scheme for anycast wireless mesh networks, which is robust to sudden traffic and network perturbations, effectively balancing load among multiple gateways and mesh nodes with little control overhead. Simulation results exhibit autonomous load balancing and failure-tolerant performance in wireless mesh networking.