Oh, I was wrong ! I though to battle someone, I had to click on a play button in the arena panel.
But it seems that actually all the listed fight have already been launch and the play buttons are just the replay buttons And the rank for each fight is given by the column with X/X like data.
So each time someone click on Fight in Area, all players rank are updated.
Will it be scalable when there will be 100000 users, wait & see
Additional info: the "select players" panel in traning mode make my chrome laggy.