Surplus Anomaly and Random Geometries

Appearance of mother universe and singular vertices in random geometries

“…Looking at the peak in the probability distribution π(n) that corresponds to the root, one can also check that it becomes smaller in inverse proportion to the polymer size, which means that independently of the size there is always the same number of roots per tree (in this case one). The detailed calculations of the vertex order distribution and the full discussion will be presented in the forthcoming publication [13].…”

“…Obviously, the situation is the same as above. In fact, the branched polymer model can be mapped exactly onto this type of balls-in-boxes model [13]. However, the balls-in-boxes model is more general because we can vary the density ρ of the balls.…”

“…However, the balls-in-boxes model is more general because we can vary the density ρ of the balls. It turns out that for fixed β the transition can also be produced by changing the density of the balls [13]. By increasing the density we can bring the system into the condensed phase (see fig.…”

Interacting Individuals Leading to Zipf's Law

“…The interesting point here is that the average size of cities is finite for α The answer is that the excess population concentrates in just one city, which therefore has a finite, large fraction of the total population. This effect has been studied recently [5] in an equilibrium model where an Hamiltonian H = i ln m i was considered. The equilibrium distribution at inverse temperature β = α is clearly given by a power law distribution.…”

Duality between equilibrium and growing networks

“…Preferential attachment [5,6] is perhaps the best known example. These two approaches are clearly different [1,4,7,8]. In the simplest example, i.e.…”