New York is filled with immigrants who can share proud tales of how they got here, but the story of how Asghar Ghahraman became Ray Alvarez is particularly improbable, and he’s gotten skilled at telling it beside the humming ice cream machines in his store. He assures listeners that everything is true, and his devoted fans do not seem to question this dramatic creation story much. The destroyer anchored off the coast of Norfolk, Va., in 1963 and Mr. Alvarez started biding his time, memorizing deck patrol schedules, until he acted on his plan. As he tells it, he leapt from the ship one night and swam two miles to shore where, soaking wet, he caught a train to Miami and slept beneath a bridge. “The alligators looked at me like I was a piece of steak,” he said. “But I had America in my pocket. That was all that mattered.”