Browsing by Author "Immonen, Johanna"
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Immonen, Johanna (2024)This thesis considers crossing probabilities in 2D critical percolation, and modular forms. In particular, I give an exposition on the theory on modular forms, percolation theory and complex analysis that is needed to characterise the crossing probabilities by means of modular forms. These results are not mine, but I review them and present full proofs which are omitted in the literature. In the special case of 2 dimensions, the percolation theory admits a lot of symmetries due to its conformal invariance at the criticality. This makes its study especially fruitful. There are various types of percolation, but let us consider for example a critical bond percolation on a square lattice. Mark each edge in the lattice black (open) or white (closed) with equal probability, and each edge independently. The probability that there is cluster of connected black edges which is attached to both left and right side of the rectangle, is the horizontal crossing probability. Note that there is always either such a black cluster connecting the left and right sides or a white cluster connecting the upper and lower sides of the rectangle in the dual lattice. This gives us a further symmetry. The crossing probability at the scaling limit, where the mesh size of the square lattice goes to zero, is given by CardySmirnov’s formula. This formula was first derived unrigorously by Cardy, but in 2001 it was proved by Smirnov in the case of a triangular site percolation. I present an alternative expression for the CardySmirnov’s formula in terms of modular forms. In particular, I show that CardySmirnov’s formula can be written as an integral of Dedekind’s eta function restricted to the positive imaginary axis. For this, one needs first that the conformal cross ratio for a rectangle corresponds to the values of the modular lambda function at the positive imaginary axis. This follows by using Schwartz reflection to the conformal map from the rectangle to the upper half plane given by Riemann mapping theorem, and finding an explicit expression for the construction using Weierstrass elliptic function. Using the change of basis for the period module and uniqueness of analytic extension, it follows that the analytic extension for the conformal cross ratio is invariant with respect to the congruent subgroup of the modular group of level 2, and is indeed the modular lambda function. Now, one may reformulate the hypergeometric differential equation satisfied by CardySmirnov’s formula as a function on lambda. Using the symmetries of lambda function, one can deduce the relation to Dedekind’s eta. Lastly, I show how CardySmirnov’s formula is uniquely characterised by two assumptions that are related to modular transformation. The first assumption arises from the symmetry of the problem, but there is not yet physical argument for the second.
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