We will discuss statistical physics in disordered systems, focusing on quantum localization and directed-polymer glass. We begin with universal fluctuations in Anderson localization and their connection to directed polymers in random media and Kardar-Parisi-Zhang universality. We then discuss Anderson transitions with the integer quantum Hall plateau transition in mind, where critical wave functions exhibit multifractality and scale-invariant fluctuations. We show that the multifractal spectrum agrees with predictions based on conformal symmetry. We next revisit the transfer-matrix formulation of directed polymers and show how products of random transfer matrices provide a unified realization of distinct one-point fluctuation laws. Finally, we turn to finite-density directed polymers, a minimal model of glassy line matter. By viewing mutually avoiding directed polymers as world lines of fermions and considering the filling of logarithmic eigenvalues of the transfer-matrix product, we obtain numerical results that agree with predictions from the replica Bethe ansatz.

Inviter: Chu-Shun Tian