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Random Permutations and Integer Partitions with Structural Constraints


Key facts

Type of research degree
Application deadline
Ongoing deadline
Country eligibility
International (open to all nationalities, including the UK)
Dr Leonid Bogachev
School of Mathematics
Research groups/institutes
<h2 class="heading hide-accessible">Summary</h2>

Permutations and integer partitions are the basic combinatorial structures that appear in numerous areas of mathematics and its applications &amp;mdash; from number theory, algebra and topology to quantum physics, statistics, population genetics, IT &amp;amp; cryptology (e.g., Alan Turing used the theory of permutations to break the Enigma code during World War II). This classic research topic dates back to Euler, Cauchy, Cayley, Lagrange, Hardy and Ramanujan. The modern statistical approach is to treat these structures as a random ensemble endowed with a suitable probability measure.

<h2 class="heading hide-accessible">Full description</h2>

<p>The uniform (equiprobable) case is well understood but more interesting models (e.g., with certain weights on the components) are mathematically more challenging. The main thrust of this PhD project is to tackle open and emerging problems about asymptotic properties of &quot;typical&quot; structures of big size, especially under certain structural constraints on the consitituent components. The focus will be on macroscopic features of the random structure, such as its limit shape.</p> <p>It is also important to study extreme values, in particular the possible emergence of a giant component which may shed light on the Bose&ndash;Einstein condensation of quantum gas, predicted in 1924 but observed only recently (Nobel Prize in Physics 2001). A related direction of research is the exploration of a deep connection with different quantum statistics; specifically, the ensemble of uniform integer partitions may be interpreted as the ideal gas of bosons (in two dimensions), whereas partitions with distinct parts correspond to fermions. In this context, an intriguing problem is to construct suitable partition classes to model the so-called anyons obeying fractional quantum statistics (also in 2D!). Furthermore, an adventurous idea may be to look for suitable partition models to mimic the unusual properties of graphene (Nobel Prize in Physics 2010), a newly discovered 2D quantum structure with certain hidden symmetries.</p> <h5>References</h5> <p>[1] Arratia, R., Barbour, A.D. and Tavar&eacute;, S. <em>Logarithmic Combinatorial Structures: a Probabilistic Approach</em>. European Math. Soc., Z&uuml;rich, 2003. (<a href="">doi:10.4171/000</a>)</p> <p>[2] Bogachev, L.V. Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts. <em>Random Structures and Algorithms</em>, <strong>47</strong> (2015), 227&ndash;266. (<a href="">doi:10.1002/rsa.20540</a>)</p> <p>[3] Bogachev, L.V. Limit shape of random convex polygonal lines: Even more universality. <em>Journal of Combinatorial Theory A</em>, <strong>127</strong> (2014), 353&ndash;399. (<a href="">doi:10.1016/j.jcta.2014.07.005</a>)</p> <p>[4] Bogachev, L.V. and Yakubovich, Yu.V. Limit shape of minimal difference partitions and fractional statistics.<em> Communications in Mathematical Physics</em>, <strong>373</strong> (2020),1085&ndash;1131. (<a href="">doi:10.1007/s00220-019-03513-5</a>)</p> <p>[5] Bogachev, L.V. and Zeindler, D. Asymptotic statistics of cycles in surrogate-spatial permutations. <em>Communications in Mathematical Physics</em>, <strong>334</strong> (2015), 39&ndash;116. (<a href="">doi:10.1007/s00220-014-2110-1</a>)</p> <p>[6] Lerda, A. <em>Anyons: Quantum Mechanics of Particles with Fractional Statistics</em>. Springer, Berlin, 1992.&gt;</p> <p>[7] Vershik, A.M. Asymptotic combinatorics and algebraic analysis. In:&nbsp;<em>Proceedings of the International Congress of Mathematicians (August 3&ndash;11, 1994 Z&uuml;rich, Switzerland)</em>, Vol. 2. Birkh&auml;user, Basel, 1995, pp. 1384&ndash;1394. (<a href=""><span id="doi-url">doi:10.1007/978-3-0348-9078-6_133</span></a>)</p> <p>The earliest start date for this project is 1 October 2020.</p>

<h2 class="heading">How to apply</h2>

<p>Formal applications for research degree study should be made online through the <a href="">University&rsquo;s website</a>. Please state clearly in the research information section&nbsp;that the research project you wish to be considered for is &lsquo;Random Permutations and Integer Partitions with Structural Constraints&rsquo; as well as&nbsp; <a href="">Dr Leonid Bogachev</a> as your proposed supervisor.</p> <p><em>We welcome applications from all suitably-qualified candidates, but UK black and minority ethnic (BME) researchers are currently under-represented in our Postgraduate Research community, and we would therefore particularly encourage applications from UK BME candidates. All scholarships will be awarded on the basis of merit.</em></p>

<h2 class="heading heading--sm">Entry requirements</h2>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline.

<h2 class="heading heading--sm">English language requirements</h2>

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid.

<h2 class="heading">Funding on offer</h2>

<p><strong>Self-funded students are also welcome to apply.</strong></p> <p><strong>UK&nbsp;students</strong>&nbsp;&ndash;&nbsp;The&nbsp;<a href="">Leeds Doctoral Scholarship (January deadline)</a>&nbsp;and the&nbsp;<a href="">William Wright Smith Scholarship&nbsp;</a>are available to UK applicants. &nbsp;<a href="">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Non-UK students</strong>&nbsp;&ndash; The&nbsp;<a href="">China Scholarship Council - University of Leeds Scholarship</a>&nbsp;is available to nationals of China. The&nbsp;<a href="">Leeds Marshall Scholarship</a>&nbsp;is available to support US citizens.&nbsp; <a href="">Alumni Bursary</a> is available to graduates of the University of Leeds.</p>

<h2 class="heading">Contact details</h2>

<p>For further information, regarding your application, please contact Doctoral College Admissions:<br /> e:&nbsp;<a href=""></a>, t: +44 (0)113 343 5057.</p> <p>For further information regarding the project, please contact Dr Leonid Bogachev by email:&nbsp;&nbsp;<a href=""></a></p>

<h3 class="heading heading--sm">Linked research areas</h3>