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I play on a server in minecraft and the best way to make money on there is to make a 7x7 chunk cactus farm because that's the servers render distance. I just recently heard about there is a 128 block radius around the player for random ticks and I can't wrap my head around what that means in context of a real farm. does that mean there is actually an 8 chunk radius around me that will grow cactus even if the server render distance is 7?

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Yes, there is. The Minecraft wiki says:

In Java Edition, chunks with a load level of 31 (see Chunk loading) or below and with centers within 128 blocks of a player are ticked on every game tick.

Source

And if I understand correctly, random ticks will still occurr in chunks, that are <render distance + 1> chunks away from you. (As stated here)

You can go ahead and expand that farm :D
Happy building and stay safe!

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