On capacity of a constrained two-dimensional channel in presence of violations

We will illustrate the connection between the Ising problem in statistical mechanics and the problem of computing the constrained capacity of an array of the same dimension. Using this connection, we show that for a given amount of violation, a soft constrained capacity can be computed. The classical Shannon capacity of a constrained channel is merely an end point of the soft capacity curve, where no violations are allowed. Moreover we reduce the problem of computing the constrained capacity to that of computing the eigenvalues of a special matrix. We claim that an analytical solution to calculating the eigenvalues of interest corresponds to solving the special case of the two-dimensional constrained channel with the constraint (1, infinity).