CRITICAL STRUCTURES · The behaviour of random discrete structures at criticality
6РП — Действия „Мария Кюри“
- Период
- 2006-10-18 → 2008-10-17
- Финансиране от ЕС
- 164 829 €
- Участници
- 1
- Схема
- EIF
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Накратко на български
Случайните дискретни структури изследват разстоянията в системи с unpredictability, като например броя връзки между уеб страници. Това помага да се разберат „фазовите преходи“, при които поведението на една система се променя рязко.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
Final Activity Report Summary - CRITICAL STRUCTURES (The behavior of random discrete structures at criticality)
How long does it take the members of a reproducing and migrating population to send its most far-flung members to the other side of the world? How far into the bedrock does groundwater seep? How many links can it take to get from one web page to another? These three seemingly distinct questions share in fact a profound similarity, since they are all about extreme distances in settings which contain randomness or unpredictability, or else random structures. Our project focussed on extreme distances, particularly on random discrete structures. Though there are many possible ways to formalise the sense in which those questions are similar, it turns out that many of these formalisations share common features. In particular, it is extremely common to see the appearance of a phase transition, which is a point at which the behaviour of the system changes drastically, e.g. from typical chains of links between websites being extremely long or nonexistent to the sudden existence of short paths between most websites. We were particularly interested in the moments at which these changes, the so-called critical phenomena, occurred, specifically as they related to distances. We made substantial progress on understanding such critical phenomena. In particular, we succeeded in establishing a metric space limit for the critical Erdos-Renyi random graph. The Erdos-Renyi random graph was one of the best-studied discrete models of random structures, nevertheless information about distances in Erdos-Renyi random graphs proved surprisingly tricky to come by. Our limit result answered a whole class of questions about typical and extreme distances in such graphs. We also answered relevant questions about the typical behaviour of smooth functions on such random graphs. These results could be seen, for example, as providing evidence on how different languages in extremely distant countries could be if pairs of neighbouring countries always had similar, but not necessarily identical, languages.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
Our primary research focus will be the behaviour of random discrete structures, with an emphasis on random graphs and percolation clusters. We will investigate the internal behaviour of such structures in the critical case - the moment when a large cluster emerges.In particular, we will investigate such questions as the cluster sizes, robustness of connectivity, and shortest paths between vertices in the same cluster. Such questions have been well-studied in both sub- and supercritical settings, but detailed information about them in the critical case remains elusive.We also aim to unify work on the heights of random trees with existing knowledge for the diameters of first-passage percolation clusters, and thereby to fully understand the moments of the diameter of first-passage percolation clusters in a wide class of infinite deterministic and random trees. We will also pursue similar questions to those listed above in random geometric graphs such as the Voronoi diagram, k-nearest-neighbour graphs, and the like.We will pursue this research using tools from combinatorics, notably random graph theory, recent developments in percolation theory such as the use of the percolation-theoretic triangle inequality in finite settings, and probabilistic tools, particularly concentration of measure inequalities. Our research is aimed at answering certain key questions that arise random discrete structures in many settings.The precise questions turn out to be slightly different depending on the setting but there is an underlying and unifying question which is driving our pursuit: what do random discrete structures look like at the moment they change from being small to being large?This research aims to be a step in elucidating that question and making some progress towards its solution.
Оригинален текст от CORDIS (на английски).
Участници
- THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD · OXFORDКоординаторОбединеното кралство
Връзки
Данни: CORDIS, © Европейски съюз