Every year, there is a lottery for permits in February. Up through this year, people would mail in their permit/lottery applications during the first two weeks in February, then wait until about April 1 to hear the results.
This process is described in the Wilderness Permit Options link (to the left), especially in box 4 on that page.
For 2012, Inyo National Forest is working with
Recreation.gov to make the lottery process a computer-based operation. We have yet to see the application process, and hope for the best with the lottery itself. If all goes as planned, the lottery will complete in minutes or hours, so people would likely know the results sometime between March 16 and April 1.
Update: Check these links for more information: