TITLE: What is the difference between hard and soft analysis?
QUESTION [16 upvotes]: I have heard references to "hard" vs. "soft" analysis. What is the difference? It seems to do with generality versus nitty-gritty estimates, but I haven't gotten any responses more clear than that.

REPLY [6 votes]: Consider the problem of shuffling a deck of cards using some shuffling technique.  One may wonder, "If I use this or that shuffling technique, will I shuffle the deck?"  The problem can be transferred to a question of whether a particular finite Markov process converges to the uniform distribution or not.  Omitting some details, a classical theorem says that, yes, the process will converge (to the uniform distribution) as long as your technique is reasonable.  Not only that, the convergence will eventually be exponential.  This seems like a useful theorem, but it is actually rather deficient.  The problem is that your practical side may wonder how many shuffles are required to get the deck reasonably randomized, and this theorem doesn't help.  So you might say that the original analysis was soft because the result does not help in solving this quantitative problem.  It tells you that shuffling is a good idea, but it doesn't give you any clue whether a given technique could shuffle the deck in your lifetime or not.  A hard analysis would tell you, for example, "If one defines reasonably randomized by measure blahblah, then $2\log_2(52)$ riffle shuffles are sufficient to randomize the deck."