Conformal Invariance of the Ising Model Interfaces and the Hausdorff Dimension of Schramm-Loewner Evolution Curves

This monograph explores the rigorous derivation of the scaling limits of planar Ising model interfaces at criticality and the subsequent determination of their fractal dimension. We review the seminal contributions of Smirnov, who established the conformal invariance of the fermionic observable, thereby proving that the scaling limit of the spin Ising interface converges to the Schramm-Loewner Evolution with parameter =3 (SLE3). Furthermore, we examine the geometric properties of these curves through the lens of Beffara's theorem, which relates the diffusivity parameter to the Hausdorff dimension of the SLE trace. Specifically, we demonstrate that the Hausdorff dimension of the critical Ising interface is almost surely dH = 11/8. The discussion encompasses the discrete complex analysis framework, the convergence of the discrete martingale observable to a holomorphic function, and the stochastic analysis underpinning the Loewner differential equation.