Superstring theoryをめぐって(放談室)

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.

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Critical phenomena in complex networks

We review recent progress on the zero-range process, a model of interacting particles which hop between the sites of a lattice with rates that depend on the occupancy of the departure site. We discuss several applications which have stimulated interest in the model such as shaken granular gases and network dynamics; we also discuss how the model may be used as a coarse-grained description of driven phase-separating systems. A useful property of the zero-range process is that the steady state has a factorized form. We show how this form enables one to analyse in detail condensation transitions, wherein a finite fraction of particles accumulate at a single site. We review condensation transitions in homogeneous and heterogeneous systems and also summarize recent progress in understanding the dynamics of condensation. We then turn to several generalizations which also, under certain specified conditions, share the property of a factorized steady state. These include several species of particles; hop rates which depend on both the departure and the destination sites; continuous masses; parallel discrete-time updating; non-conservation of particles and sites.

https://iopscience.iop.org/article/10.1088/0305-4470/38/19/R01/pdf

Nonequilibrium statistical mechanics of the zero-range process and related models

Nonperturbative quantum gravity

Causal Dynamical Triangulations in four dimensions provide a background-
independent definition of the sum over geometries in nonperturbative quantum
gravity, with a positive cosmological constant. We present evidence that a macro-
scopic four-dimensional world emerges from this theory dynamically.

P. Bialas

Papers

Condensation in the Backgammon model

Focusing on the fixed point of 4D simplicial gravity

Appearance of mother universe and singular vertices in random geometries

Phase diagram of the mean field model of simplicial gravity

Phase transition in fluctuating branched geometry