No, it has nothing to do with where it's concentrated; I don't know of any tumbler designs that aren't concentrated in one place.
It has to do with that if Alice gets a Bitcoin by selling uncensored newspapers in China, transfers it to wallet A in the tumbler, and then the tumbler transfers that to Alice's Coinbase account, Alice is going to jail. The transaction record tells us that Alice's Bitcoin came from the uncensored newspaper through an intermediary. The Chinese government can then arrest Alice on suspicion of selling illegal newspapers. At that point Alice must either prove somehow that wallet A in the tumbler is owned by the person who really sold the newspapers (which will be impossible), or admit that she sold the newspapers and used a (bad) tumbler in an attempt to hide her transaction. That is to say, if there's a chain of transactions connecting Alice to the newspaper sale, the tumbler has provided Alice no anonymity.
Contrast this with a more realistic tumbler design: the tumbler owner (let's call him Troy) deposits a bitcoin from Coinbase in wallet A. Alice is the first user of the tumbler, so she gets a bitcoin selling uncensored newspapers in China and deposits it in wallet B. The tumbler owner then forwards the coin from wallet A (minus a fee) into Alice's Coinbase account. Now Bob, as the second user of the tumbler, comes along and deposits a coin he got from selling LSD on the Silk Road into wallet C. Troy then sends the coin Alice got from selling uncensored newspapers (minus a fee) to Bob's Coinbase account.
Alice now has a coin from Coinbase, Bob has a coin from Alice's sale of an uncensored newspaper in China, and Troy the tumbler operator has a coin sitting in wallet C waiting for the next person to use the tumbler. The Chinese government has no way to track the transactions to Alice. IF Bob lives in China the Chinese government could come and accuse him of selling uncensored newspapers, but the accusation would make no sense; he has alibis for some of the days newspapers were sold, he's not educated enough to write, etc., because he didn't sell the newspapers. Bob can even say he used a tumbler, but he used it because he purchased a lot of bondage pornography: embarrassing, but not illegal. But more likely than not, Bob won't even be accused of Alice's crimes, either because he lives in a different country or because the accusations won't make any sense. Both Alice and Bob's anonymity has been protected.
Great explanation, I understand it better from a technical perspective.
I assume Troy is hidden to the authorities, who only have the Blockchain ledger and no other information to go on? That's waht I meant by "concentrated" above -- how likely is it that the Chinese government would find Troy and extract the information out of him with legal threats or with a heavy wrench?
I don't think the owner of the tumbler has any motivation to keep records (after each transaction is complete). So he deletes the records and spares himself the heavy wrench.
If Troy is trustworthy, he'll delete the transaction history.
However, there area number of attacks possible from Troy's perspective. He can keep a transaction history and use it to blackmail Alice and Bob. He can simply keep coins sent to the tumbler and send out nothing. It's very important that Troy be a trusted actor.
> IF Bob lives in China the Chinese government could come and accuse him of selling uncensored newspapers [...] Bob can even say he used a tumbler, but he used it because he purchased a lot of bondage pornography: embarrassing, but not illegal.
Not relevant to the example, but this is totally illegal.
It has to do with that if Alice gets a Bitcoin by selling uncensored newspapers in China, transfers it to wallet A in the tumbler, and then the tumbler transfers that to Alice's Coinbase account, Alice is going to jail. The transaction record tells us that Alice's Bitcoin came from the uncensored newspaper through an intermediary. The Chinese government can then arrest Alice on suspicion of selling illegal newspapers. At that point Alice must either prove somehow that wallet A in the tumbler is owned by the person who really sold the newspapers (which will be impossible), or admit that she sold the newspapers and used a (bad) tumbler in an attempt to hide her transaction. That is to say, if there's a chain of transactions connecting Alice to the newspaper sale, the tumbler has provided Alice no anonymity.
Contrast this with a more realistic tumbler design: the tumbler owner (let's call him Troy) deposits a bitcoin from Coinbase in wallet A. Alice is the first user of the tumbler, so she gets a bitcoin selling uncensored newspapers in China and deposits it in wallet B. The tumbler owner then forwards the coin from wallet A (minus a fee) into Alice's Coinbase account. Now Bob, as the second user of the tumbler, comes along and deposits a coin he got from selling LSD on the Silk Road into wallet C. Troy then sends the coin Alice got from selling uncensored newspapers (minus a fee) to Bob's Coinbase account.
Alice now has a coin from Coinbase, Bob has a coin from Alice's sale of an uncensored newspaper in China, and Troy the tumbler operator has a coin sitting in wallet C waiting for the next person to use the tumbler. The Chinese government has no way to track the transactions to Alice. IF Bob lives in China the Chinese government could come and accuse him of selling uncensored newspapers, but the accusation would make no sense; he has alibis for some of the days newspapers were sold, he's not educated enough to write, etc., because he didn't sell the newspapers. Bob can even say he used a tumbler, but he used it because he purchased a lot of bondage pornography: embarrassing, but not illegal. But more likely than not, Bob won't even be accused of Alice's crimes, either because he lives in a different country or because the accusations won't make any sense. Both Alice and Bob's anonymity has been protected.