I believe that only off the screen, in the standard spawn range, I ended up doing some tests now in the mobile version:
There are two logs on my map, one near a village and the other isolated, in both I placed a water candle and a gnome statue near, however, in the second I made a "box" around it 37 blocks high and 9 blocks wide, so the fairies don't run away, the reason for being 37 tall is because of the spawn range mentioned above, I left the player on top of the box with minions killing the mobs outside, and i no had any fairies, even during the night, in the second test I went into space and came back to the trunk to unload and load the spawn in the area, and the result was same.
In the log near the village I didn't change anything, I just kept moving away and getting closer during the night, and I ended up finding a single fairy hovering over him.
I think I can conclude that fairies are like critters, as they can be influenced by water candles and nearby NPCs / Villages, so if you want an efficient farm, I suggest blocking all spawn around the Log, building a village nearby and capture / kill all the critters that appear nearby to make room for the fairies.