After recalling the concept of the Hausdorff dimension, we study the fractal properties of a quantum particle path. As a novelty we consider the possibility for the space where the particle propagates to be endowed with a quantum-gravity-induced minimal length. We show that the Hausdorff dimension accounts for both the quantum mechanics uncertainty and manifold fluctuations. In addition the presence of a minimal length breaks the self-similarity property of the erratic path of the quantum particle. Finally we establish a universal property of the Hausdorff dimension as well as the spectral dimension: They both depend on the amount of resolution loss which affects both the path and the manifold when quantum gravity fluctuations occur.Comment: 7 pages, 4 figure, updated version which matches that published on Physical Review

We compute the partition function of the q-states Potts model on a random planar lattice with p ≤ q allowed, equally weighted colours on a connected boundary. To this end, we employ its matrix model representation in the planar limit, generalising a result by Voiculescu for the addition of random matrices to a situation beyond free probability theory. We show that the partition functions with p and q − p colours on the boundary are related algebraically. Finally, we investigate the phase diagram of the model when 0 ≤ q ≤ 4 and comment on the conformal field theory description of the critical points.Submitted to: J. Phys. A: Math. Theor.

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The resolvent operator plays a central role in matrix models. For instance, with utilizing the loop equation, all of the perturbative amplitudes including correlators, the free-energy and those of instanton corrections can be obtained from the spectral curve of the resolvent operator. However, at the level of non-perturbative completion, the resolvent operator is generally not sufficient to recover all the information from the loop equations. Therefore it is necessary to find a sufficient set of operators which provide the missing non-perturbative information. In this paper, we study generalized Wronskians of the Baker-Akhiezer systems as a manifestation of these new degrees of freedom. In particular, we derive their isomonodromy systems and then extend several spectral dualities to these systems. In addition, we discuss how these Wronskian operators are naturally aligned on the Kac table. Since they are consistent with the Seiberg-Shih relation, we propose that these new degrees of freedom can be identified as FZZT-Cardy branes in Liouville theory. This means that FZZT-Cardy branes are the bound states of elemental FZZT branes (i.e. the twisted fermions) rather than the bound states of principal FZZT-brane (i.e. the resolvent operator).Comment: 131 pages, 4 figure