EPSRC Reference: 
EP/P003656/1 
Title: 
Random Walks and Quantum Spin Systems 
Principal Investigator: 
Toth, Professor B 
Other Investigators: 

Researcher CoInvestigators: 

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: 

Summary on Grant Application Form 
Random motions with long memory:
Markovian random walks (with short memory) in ddimensional 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 longtime 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 diffusionlike 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 selfinteractions, 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 longtime asymptotic scaling behaviour of these processes. We will study the longtime asymptotics (socalled 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 socalled offdiagonal long range order  a kind of magnetic ordering  in particular instances. The relevance of the problem is emphasized by noting that offdiagonal long range order is equivalent to BoseEinstein 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 offdiagonal longrange 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 FeynmanKac 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 spin1/2 isotropic quantum Heisenberg model with ferromagnetic couplings, reformulate the infrared bound as a probabilistic correlation inequality for a particular stochastic interacting particle system (the socalled symmetric simple exclusion model) and prove it this way.

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Organisation Website: 
http://www.bris.ac.uk 