New percolation crossing formulas and second-order modular forms (Q2638182)

The crossing probabilities of of a rectangular percolation grid (or the limit thereof, as the grid becomes finer) is a function of the aspect ration \(r=\text{width}/\text{height}\). One considers the horizontal crossing probability \(\text{P}_h(r)\), which is the probability of at least one cluster connecting the left and right sides of the grid, and the horizontal-vertical probability \(\text{P}_{hv}(r)\), i.e., the probability of the existence of a cluster connecting all four sides. Explicit expressions for these probabilities have been found by physical arguments and have been proved rigorously by \textit{J. Dubédat} [Probab. Theory Relat. Fields 134, No.~3, 453--488 (2006; Zbl 1112.60032)] and \textit{S. Smirnov} [C. R. Acad. Sci., Paris Sér. I, Math. 333, No.~3, 239--244 (2001; Zbl 0985.60090)]. Based on these expressions, \textit{P. Kleban} and \textit{D. Zagier} found [J. Stat. Phys. 113, No.~3--4, 431--454 (2003; Zbl 1081.60067)] an interesting modular behavior of these probability functions. Recently, \textit{J. J. H. Simmons, P. Kleban} and \textit{R. M. Ziff} [J. Phys. A, Math. Theor. 40, No.~31, F771--F784 (2007; Zbl 1147.82335)] have found similar explicit formulae for related but different crossing probabilities. The current paper interprets these as weakly holomorphic second order modular forms of weight 0 on the congruence group \(\Gamma(2)\). It is shown that, under additional assumptions, the crossing probabilities are completely determined by their transformation laws.