Strings, Matrix Models, and Meanders

TL;DR

This paper reviews the current understanding of bosonic strings and discretized random surfaces, explores the Kazakov-Migdal model with a logarithmic potential, and discusses the meander problem's matrix model representations and their connections.

Contribution

It introduces new matrix model representations for the meander problem and relates them to the Kazakov-Migdal model, including supersymmetric approaches.

Findings

Exact solutions for the Kazakov-Migdal model at large D.

New matrix model formulations for the meander problem.

Insights into the phase structure of discretized random surfaces.

Abstract

I briefly review the present status of bosonic strings and discretized random surfaces in D>1 which seem to be in a polymer rather than stringy phase. As an explicit example of what happens, I consider the Kazakov-Migdal model with a logarithmic potential which is exactly solvable for any D (at large D for an arbitrary potential). I discuss also the meander problem and report some new results on its representation via matrix models and the relation to the Kazakov-Migdal model. A supersymmetric matrix model is especially useful for describing the principal meanders.