| EPSRC Reference: |
EP/K01899X/1 |
| Title: |
Whittaker processes and stochastic integrable systems |
| Principal Investigator: |
O'Connell, Professor NM |
| Other Investigators: |
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| Researcher Co-Investigators: |
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| Project Partners: |
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| Department: |
Mathematics |
| Organisation: |
University of Warwick |
| Scheme: |
EPSRC Fellowship |
| Starts: |
01 April 2013 |
Ends: |
31 March 2018 |
Value (£): |
936,394
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| EPSRC Research Topic Classifications: |
| Algebra & Geometry |
Mathematical Analysis |
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| EPSRC Industrial Sector Classifications: |
| No relevance to Underpinning Sectors |
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| Panel History: |
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Summary on Grant Application Form |
The proposed research lies at the interface of probability, integrable systems and representation theory. Central to this research program are the Whittaker measures and related processes introduced in my recent papers. These remarkable stochastic processes arise at the intersection of several different areas of mathematics which include: crystals and path models in representation theory and their geometric counterparts; totally positive varieties; `tropical RSK correspondence'; random polymers and surface growth; the Kardar-Parisi-Zhang (KPZ) equation; classical and quantum integrable systems; random matrix theory. The proposed research will make novel contributions in, and draw connections between, all of these areas. As such it is a highly intra-disciplinary research program. It is also very timely. There has been an explosion of activity in this area in the last year or so which has attracted widespread attention.
In recent years, with my collaborators, I have been developing probabilistic constructions related to Littlemann's path model in representation theory, where roughly speaking the combinatorial models of representation theory are replaced by continuous models and instead of enumerating paths on a lattice we integrate with respect to Wiener measure on continuous paths. In the context of $GL(n)$, our results are related to the recently discovered and celebrated connections between increasing subsequences, directed percolation and random matrices and provide a unified way of understanding these connections; they can also be understood as a far-reaching generalisation of Pitman's celebrated `$2M-X$' theorem, a classical result which gives a deep connection between one-dimensional Brownian motion and the three-dimensional Bessel process.
The above path models are defined by expressions in the so-called max-plus algebra. I recently discovered that in the context of $GL(n)$ a geometric lifting of these constructions (that is, replace all of these expressions by the corresponding expressions in the usual algebra) gives rise to a remarkable connection with the open quantum Toda chain with $n$ particles. At the centre of this connection are the Whittaker measures and processes referred to above. These are closely related to $GL(n)$-Whittaker functions which are eigenfunctions of the open quantum Toda chain. In the $GL(n)$ setting, the geometrically-lifted path model can be thought of as a continuous-time version of the `tropical RSK correspondence' introduced by A.N. Kirillov (2000). We have recently established a direct connection between the tropical RSK correspondence and $GL(n)$-Whittaker functions which, in particular, gives an interpretation of the Bump-Stade Whittaker integral identity as an analogue of the Cauchy-Littlewood identity for tropical RSK.
These results have important applications to random polymers. In the mathematical physics interpretation, the geometric lifting (e.g. from classical RSK to `tropical RSK') corresponds to the passage from zero to positive temperature. As such, these developments can also be regarded as a positive temperature extension of random matrix theory---for example, special cases of the Whittaker measures introduced in my recent papers can be seen as extensions from zero to positive temperature of the Gaussian and Laguerre Unitary ensembles. The integrable structure discovered in these works also provides a new approach towards understanding the apparent integrability of the KPZ (or stochastic heat) equation, which is still somewhat of a mystery.
These recent developments reveal a fascinating interplay between classical, quantum and stochastic integrable systems which is main theme of this research programme.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
| Summary |
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| Date Materialised |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.warwick.ac.uk |