Graphical representations of classical and quantum spin systems

This project will explore connections between models of Statistical physics (The Bose gas, the quantum and classical (anti-)ferromagnet, the Ising model), related probabilistic models (the random path model, the random interchange/stirring model, the random current model) and some "complex" spin systems (that is - spin systems where spins take complex values).<br /> <br /> The aim will be to understand properties of these models such as phase transitions, percolation, decay of correlations by using the connections between the three familes of model and hence learn more about the models than would be possible by looking at only one of the three families.

<p>Many models from Statistical Physics, such as the Bose gas, (quantum and classical) ferromagnet/anti-ferromagnet, and the Ising model, have probabilistic representations in terms of one-dimensional object (Brownian or random walk paths, loops, dimers….) living in the same space as the physical model. This space could be a graph such as a regular lattice, or a subset of the familiar three-dimensional reals. It has recently been noticed that they can also be representation as a different kind of spin system where spins take complex values.</p> <p>These representations have many attractive properties for Mathematicians. They are often quite beautiful probabilistic models for study in their own right but also give the opportunity to learn about important models from Physics with a new array of tools. They do however come with challenges, many of these models have complicated correlations that means there is no (or very little) independence in the model. This difficulty can sometimes be overcome, but even when it cannot there are tools available to us.</p> <p>This project will explore these models both as probabilistic objects isolated from Physics and as physical models. Questions of percolation and other phase transitions in these models will be explored, as well as decay of correlations. This will be achieved by using the complex spin and probabilistic representations which allows us to switch between different techniques and approaches.</p>

<p style="text-align:start">Formal applications for research degree study should be made online through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University's website</a>.Please state clearly in the Planned Course of Study section that you are applying for <em><strong>PHD Statistics FT</strong></em> and in the research information section that the research degree you wish to be considered for is <em><strong>Combinatorial identities from interacting particle systems</strong></em> as well as <a href="https://eps.leeds.ac.uk/faculty-engineering-physical-sciences/staff/13180/dr-ben-lees">Dr. Benjamin Lees</a> as your proposed supervisor and in the finance section, please state clearly <em><strong>the funding that you are applying for, if you are self-funding or externally sponsored (including the name of your sponsor)</strong></em>.</p> <p>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <p>Applications will be considered on an ongoing basis.  Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application.  We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed.</p> <p><strong>If you are applying for University or School Scholarships for 2026/27 entry, with external sponsorship or you are funding your own study, please ensure you provide your supporting documents at the point you submit your application:</strong></p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date including the grading scheme</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul> <p><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p>

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. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

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. Some schools and faculties have a higher requirement.

<p><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>Scholarship opportunities open from October 2025</strong></p> <p><strong>UK</strong> – The <a href="https://phd.leeds.ac.uk/funding/138-leeds-doctoral-scholarship-2025-faculty-of-engineering-and-physical-sciences#:~:text=Key%20facts&text=One%20Leeds%20Doctoral%20Scholarship%20is,rata%20for%20part%2Dtime%20study.">Leeds Doctoral Scholarship</a> <strong>(closing date: February 2026)</strong> and <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> <strong>(closing date: April 2026)</strong> are available to UK applicants.  <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Non-UK</strong> – The <a href="https://phd.leeds.ac.uk/funding/48-china-scholarship-council-university-of-leeds-scholarships-2021">China Scholarship Council - University of Leeds Scholarship</a> is available to nationals of China <strong>(closing date: January 2026)</strong>. The <a href="https://phd.leeds.ac.uk/funding/73-leeds-marshall-scholarship">Leeds Marshall Scholarship</a> is available to support US citizens. <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p>Please note that if you are successful in securing an academic offer for PhD study, this does not mean that you have been successful in securing an offer of funding.</p> <p>If you are applying for the Leeds Doctoral Scholarship, Leeds Opportunity Research Scholarship, China Scholarship Council-University of Leeds Scholarship or Leeds Marshall Scholarship, you will need to complete a separate application, specific to these scholarships, to be considered for funding.</p> <p>You will be responsible for paying the overtime fee in full in your writing up/overtime year (£340 in Session 2025/26), but the scholarship maintenance allowance will continue to be paid for up to 6 months in the final year of award.</p> <p><strong>Important: </strong>Please note that that the award does <em><strong>not</strong></em> cover the costs associated with moving to the UK.  All such costs (<a href="https://www.leeds.ac.uk/international-visas-immigration/doc/applying-student-visa">visa, Immigration Health Surcharge</a>, flights etc) would have to be met by yourself, or you will need to find an alternative funding source. </p> <p>Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p>

<p style="text-align:start">For further information about this project, please contact Dr Benjamin Lees by email to <a href="mailto:b.t.lees@leeds.ac.uk">b.t.lees@leeds.ac.uk</a></p> <p>For further information about your application, please contact PGR Admissions by email to <a href="mailto:maps.pgr.admissions@leeds.ac.uk">maps.pgr.admissions@leeds.ac.uk</a></p> <p> </p>