𝔖 Bobbio Scriptorium
✦   LIBER   ✦

The phase transition in the evolution of random digraphs

✍ Scribed by Tomasz Łuczak

Publisher
John Wiley and Sons
Year
1990
Tongue
English
Weight
271 KB
Volume
14
Category
Article
ISSN
0364-9024

No coin nor oath required. For personal study only.

✦ Synopsis

Abstract

Let \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {\rm D}\limits^ \to $\end{document}(n, M) denote a digraph chosen at random from the family of all digraphs on n vertices with M arcs. We shall prove that if M/n ≤ c < 1 and ω(n) → ∞, then with probability tending to 1 as n → ∞ all components of \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {\rm D}\limits^ \to $\end{document}(n, M) are smaller than ω(n), whereas when M/n ≥ c > 1 the largest component of \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {\rm D}\limits^ \to $\end{document}(n, M) is of the order n with probability 1 ‐ o(1).

📜 SIMILAR VOLUMES

Phase Transitions in the Evolution of Pa
Phase Transitions in the Evolution of Partial Orders
✍ Hans Jürgen Prömel; Angelika Steger; Anusch Taraz 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 331 KB

We determine the approximate number of partial orders with a fixed number of comparable pairs, give a complete description of the evolution of partial orders, and prove that infinitely many phase transitions occur. This answers questions posed by Dhar, Kleitman, and Rothschild 20 years ago.

The major transitions in evolution
The major transitions in evolution
✍ Arnold I. Miller 📂 Article 📅 1997 🏛 John Wiley and Sons 🌐 English ⚖ 48 KB

diverse. The process of hypervariable plasticity followed by selection is as common as it is inelegant; it is so simple and powerful a scheme that real-world organisms have evolved versions of it over and over again.

A Phase Transition for the Minimum Free
A Phase Transition for the Minimum Free Energy of Secondary Structures of a Random RNA
✍ Momiao Xiong; Michael S. Waterman 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 209 KB

The free energy of a single-stranded RNA can be calculated by adding the free energies of the components: basepairs, bulges, and loops. Basepairs receive negative free energy while the unpaired bases receive positive free energy. The minimum free energy of a random RNA secondary structure with one d

The Lineshape of Saturation-Broadened Tr
The Lineshape of Saturation-Broadened Transitions in the Gas Phase
✍ R.H. Schwendeman 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 84 KB

The equations of motion for the density matrix are manipulated in spherical tensor form to derive formulas for the lineshapes of saturation-broadened spectroscopic transitions for gas-phase molecules. The formulas, which include consideration of the spatial degeneracy, are given for both plane-polar