It's important to realize that areas we think of as arid now were not always deserts. For example, take Mesopotamia. The sea used to be closer to Mesopotamia, providing a marshland. There were also human disasters, such as when a king diverted a river in a failed attempt to quell his restless people and couldn't put it back. There was also over farming, which made the land salty. (Sources: Van de Meiroop, "Society and Enterprise in Old Babylonian Ur 1992, Van de Meiroop "a History of the Ancient Near East")

As the other commenter states, the Persian empire was not necessarily arid. However, you are correct in thinking that they had a centralized redistribution system. They would extract resources as tribute/trade from an area rich in that particular resource and then redistribute it from a central hub to areas in need. There are reliefs at Persepolis that are thought to maybe represent this ritual. (Allen, Lindsay "The Persian Empire" 2005)

Can I ask for clarification here: I always thought that the Persian empire was fertile rather than arid. I just looked it up and it included the fertile crescent, the Nile valley and much of modern day Iran and Turkey, both of which have a high agricultural output now.

Most of what needs to be said has already been said. But Persia was incredibly large, and mostly centered around Modern Day Iran/Iraq. Iraq had two massive rivers running through the center of the country. There is certainly desert in what is now Western Iraq, and Southern Syria but those werent heavily populated (and still arent to this day). Most cities were around water, and thus farming. That said, there was massive trade throughout Persia.

For example, major cities through different Eras, Babylon, Hatra, Ctesphon, Persepolis, Baghdad were all by fairly major bodies of water and were rich in farmland.