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Details of Grant 

EPSRC Reference: EP/P003656/1
Title: Random Walks and Quantum Spin Systems
Principal Investigator: Toth, Professor B
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics
Organisation: University of Bristol
Scheme: EPSRC Fellowship
Starts: 01 October 2016 Ends: 31 July 2022 Value (£): 662,565
EPSRC Research Topic Classifications:
Statistics & Appl. Probability
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
19 Jul 2016 EPSRC Mathematical Sciences Fellowship Interviews July 2016 Announced
08 Jun 2016 EPSRC Mathematics Prioritisation Panel Meeting June 2016 Announced
Summary on Grant Application Form
Random motions with long memory:

Markovian random walks (with short memory) in d-dimensional space have been studied since the early years of the twentieth century as simple models of various phenomena (like diffusion of particles suspended in fluids in thermal equilibrium). By now all aspects of these random processes are fully understood. However, true physical (and other natural) phenomena are much more complicated than these naïve models and the methods of classical probability simply do not suffice for their study. In particular, due to interactions of the observed moving particle with its environment long-time correlations build up and therefore the naïve Markovian approximations become unsuitable. Since the early 1980s very intense research has been concentrated on understanding more realistic mathematical models of diffusion-like phenomena. Among the most investigated classes of models are the following:

-- Random walks and diffusions in random environment, where the local rules of the random walker are spatially inhomogeneous and randomly sampled themselves.

-- Random walks with self-interactions, where the random walker's local rules are influenced by its own past trajectory via some local functional of its own occupation time measure.

-- Diffusion under deterministic (typically Hamiltonian) dynamics, where the randomness comes only with the initial conditions of the system, e.g. due to thermal equilibrium.

Our research ambition is to understand the long-time asymptotic scaling behaviour of these processes. We will study the long-time asymptotics (so-called scaling limits) of these processes, proving normal or anomalous diffusion in relevant models.

Stochastic representations for quantum spin systems:

More than eighty years since its formulation the quantum Heisenberg model of interacting spins is still a fundamental model of quantum statistical physics. It is sufficiently rich to encode complex physical phenomena and pose deep mathematical challenges. The main problem is the existence of so-called off-diagonal long range order - a kind of magnetic ordering - in particular instances. The relevance of the problem is emphasized by noting that off-diagonal long range order is equivalent to Bose-Einstein condensation, thus being of paramount importance in understanding of superfluidity. In a ground breaking work published in 1978 Dyson, Lieb and Simon (DLS) proved the occurrence of off-diagonal long-range order at low positive temperatures in 3 dimensions, for models with antiferromagnetic interactions. Since then a similar result for ferromagnetic couplings escapes all attempts of rigorous mathematical proof. The main problem here is establishing a particular correlation inequality called infrared bound. The method of DLS substantially relies on a particular feature of the antiferromagnetic models, called reflection positivity, which simply doesn't hold in ferromagnetic cases. Nevertheless, the infrared bound is expected to hold. A major challenge for the specialists working in this field is to find some way around the reflection positivity argument and arrive at the truly relevant correlation inequalities (infrared bounds) by other means.

Stochastic representations of the quantum spin systems arise via a beautiful link with probability, the Feynman-Kac formula, and recently became rather popular, since they lead to probabilistic reformulations of the relevant quantum statistical physics problems, typically in terms of interacting stochastic particle systems or stochastic geometric objects, like random loops on graphs.

My main ambition in this context is to apply a well suited probabilistic reformulation of the spin-1/2 isotropic quantum Heisenberg model with ferromagnetic couplings, reformulate the infrared bound as a probabilistic correlation inequality for a particular stochastic interacting particle system (the so-called symmetric simple exclusion model) and prove it this way.
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