This is a bit unusual post in the “test your intuition” corner as the problem is not entirely formal.
How does random noise in the digital world typically look?
Suppose you have a memory of n bits, or a memory based on a larger r-letters alphabet, and suppose that a “random noise” hits the memory in such a way that the probability of each bit being affected is t.
What will be the typical behavior of such a random digital noise? Part of the question is to define “random noise” in the best way possible, and then answer it for this definition.
In particular, Will the “random noise” typically behave in a way which is close to be independent on the different bits? or will it be highly correlated? or pehaps the answer depends on the size of the alphabet and the value of t?
The source of this question is an easy fact about quantum memory which asserts that if you consider a random noise operation acting on a quantum memory with n qubits, and if the probability that every qubit is damaged is a tiny real number t, then typically the noise has the following form: with large probability nothing happens and with tiny probability (a constant times t) a large fraction of qubits are harmed.
I made one try for the digital (binary) question but I am not sure at all that it is the “correct” definition for what “random noise” is.
Update: over “mathoverflow” Greg Kuperberg made an appealing argument that for the correct notion of random noise the behavior in the classical case is similar to that of the quantum case.