This paper is dedicated to the 70th birthday of V. Arnol’d. The paper is motivated by emerging connections between the conformal field theory (CFT) on the one hand and stochastic Löwner evolution (SLE) processes and measures that play the role of the Haar measures for the diffeomorphism group of a circle, on the other hand. We attempt to build a framework for widely spread beliefs that SLE processes would provide a picture of phase separation in a small massive perturbation of the CFT.

The paper gives a review of three case studies of complexity of production systems in manufacturing and commercial industry and develops mathematical methods stemming from these studies. We use as measures of complexity various (long term) entropy rates that naturally emerge in the analysis of systems under consideration; in our case, the main focus is on (physical or virtual) queues and related phenomena. Consequently, a system is considered ‘more complex’ when its entropy rates are higher. The same principle is applied when different subsystems of a given system are compared with each other, identifying a ‘bottleneck’. The numerical values for entropy rates are determined in the course of observation and recording, subject to some simplifying assumptions. To enable us to make effective comparisons, we introduce various classifications of queue-related conditions in systems under investigation. We also discuss a number of practical issues that emerge here, including noise and data loss.

We propose a class of models of random walks in a random environment where an exact solution can be given for a stationary distribution. The environment is cast in terms of a Jackson/Gordon–Newell network although alternative interpretations are possible. The main tool is the detailed balance equations. The difference compared to earlier works is that the position of the random walk influences the transition intensities of the network environment and vice versa, creating strong correlations. The form of the stationary distribution is closely related to the well-known product formula.