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We consider the problem of neighbor discovery in static wireless ad hoc networks with directional antennas. We propose several probabilistic algorithms in which nodes perform random, independent transmissions to discover their one-hop neighbors. Our neighbor discovery algorithms are classified into two groups, viz. Direct-Discovery Algorithms in which nodes discover their neighbors only upon receiving a transmission from their neighbors and Gossip-based algorithms in which nodes gossip about their neighbors' location information to enable faster discovery. We first consider the operation of these algorithms in a slotted, synchronous system and mathematically derive their optimal parameter settings. We show how to extend these algorithms for an asynchronous system and describe their optimal design. Analysis and simulation of the algorithms show that nodes discover their neighbors much faster using gossip-based algorithms than using direct-discovery algorithms. Furthermore, the performance of gossip-based algorithms is insensitive to an increase in node density. The efficiency of a neighbor discovery algorithm also depends on the choice of antenna beamwidth. We discuss in detail how the choice of beamwidth impacts the performance of the discovery process and provide insights into how nodes can configure their beamwidths.