Random Graphs, Spin Glasses and Percolation

Given a function f : [n] → {0, 1}, its k-wise direct product encoding is the function F : [n]k → {0, 1}k defined by F (x1, . . . , xk) = (f(x1), . . . , f(xk)). This simple “encoding” is useful in PCP-like settings, when we want to simulate reading k arbitrary values of f by accessing only a constant number of inputs of the encoding F . The main challenge is to test that a given F is indeed a ”correct” encoding of some f by reading only few (ultimately 2) random values in F . This is called a local consistency test. In the talk I will survey some variants of the problem and discuss the context in which this problem arises, with particular focus on gap amplification.