a correlation function, namely the spin-spin correlation function: where s is the spin quantum number and x is the distance function within the particular system. This function is equal to unity when and decreases as the distance increases. Typically, it decays exponentially to zero at large distances, and the system is considered to be disordered. But if the correlation function decays to a constant value at large then the system is said to possess long-range order. If it decays to zero as a power of the distance then it is called quasi-long-range order (for details see Chapter 11 in the textbook cited below. See also Berezinskii–Kosterlitz–Thouless transition). Note that what constitutes a large value of is understood in the sense of asymptotics. Quenched disorder In statistical physics, a system is said to present quenched disorder when some parameters defining its behavior are random variables which do not evolve with time, i.e. they are quenched or frozen. Spin glasses are a typical example. It is opposite to annealed disorder, where the random variables are allowed to evolve themselves. In mathematical terms, quenched disorder is harder to analyze than its annealed counterpart, since the thermal and the noise averaging play very different roles. In fact, the problem is so hard that few techniques to approach each are known, most of them relying on approximations. The most used are a technique based on a mathematical analytical continuation known as the replica trick the Cavity method; although these give results in accord with experiments in a large range of problems, they are not generally proven to be a rigorous mathematical procedure. More recently it has been shown by rigorous methods, however, that at least in the archetypal spin-glass model (the so-called Sherrington–Kirkpatrick model) the replica based solution is indeed exact. The second most used technique in this field is generating functional analysis. This method is based on path integrals, and is in principle fully exact, although generally more difficult to apply than the replica procedure. Annealed disorder A system is said to present annealed disorder when some parameters entering its definition are random variables, but whose evolution is related to that of the degrees of freedom defining the system. It is defined in opposition to quenched disorder, where the random variables may not change their values. Systems with annealed disorder are usually considered to be easier to deal with mathematically, since the average on