Statistics of geometric clusters in Potts model: statistical mechanics approach

The percolation of Potts spins with equal values in Potts model on graphs (networks) is considered. The general method for finding the Potts clusters' size distributions is developed. It allows full description of percolation transition when a giant cluster of equal-valued Potts spins appears. The method is applied to the short-ranged q-state ferromagnetic Potts model on the Bethe lattices with the arbitrary coordination number z . The analytical results for the field-temperature percolation phase diagram of geometric spin clusters and their size distribution are obtained. The last appears to be proportional to that of the classical non-correlated bond percolation with the bond probability, which depends on temperature and Potts model parameters.

The existence of a giant cluster for percolation on large Crump–Mode–Jagers trees

In this paper we consider random trees associated with the genealogy of Crump–Mode–Jagers processes and perform Bernoulli bond-percolation whose parameter depends on the size of the tree. Our purpose is to show the existence of a giant percolation cluster for appropriate regimes as the size grows. We stress that the family trees of Crump–Mode–Jagers processes include random recursive trees, preferential attachment trees, binary search trees for which this question has been answered by Bertoin [7], as well as (more general) m-ary search trees, fragmentation trees, and median-of-( $2\ell+1$ ) binary search trees, to name a few, where to our knowledge percolation has not yet been studied.

Advancing Sustainability Research in the 21st Century

Major developments have been achieved in advancing sustainability research in sustainability science (SS) as a new and distinct field. Based on its emerging academic landscape, SS has now acquired an unusual geographic footprint with a giant cluster of co-authorships in cities and countries at very different levels, all in pursuit of sustainable development and sustainability. It has gained a robust inclusion in the agendas of governments and corporations along with the burgeoning growth of educational and research programs worldwide. In the call for the integration of research and education, higher educational and research institutions have fostered the goals toward sustainability to address the practical protection of Earth's key support systems, climate change, and other global issues. In confronting a wide range of global issues in the 21st century, scientists across the globe see the need for accelerating knowledge in sustainability wherein knowledge production itself must be made sustainable.