Details of Grant 

EPSRC Reference:	EP/M027694/1
Title:	Continuous gradient interfaces with disorder
Principal Investigator:	Cotar, Professor C
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Department:	Statistical Science
Organisation:	UCL
Scheme:	First Grant - Revised 2009
Starts:	15 November 2015	Ends:	14 May 2017	Value (£):	98,386
EPSRC Research Topic Classifications:	Mathematical Analysis Statistics & Appl. Probability
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 03 Mar 2015 EPSRC Mathematics Prioritisation Panel March 2015 Announced
Summary on Grant Application Form
Continuous gradient models are natural generalizations to higher d-dimensional time of the standard random walk and have drawn a lot of attention lately. Partly, this is due to the fact that the contour lines of their interface height converge in d=2 to Schramm's SLE - a family of random planar curves shown to be the universal scaling limit of many important two-dimensional lattice models in probability and statistical mechanics (2006 Fields Medal for Werner). Moreover, gradient models are connected to random interlacements, a novel probability area pioneered by Sznitman, to reinforced random walks, and to Liouville quantum gravity. Informally, the random interface is given by highly-dependent real-valued random variables whose distribution is a function of the nearest-neighbour interactions V of the interface. In the case with V a quadratic function, this distribution is a Gaussian measure - the Gaussian Free Field (GFF) - the d-dimensional time analog of Brownian motion. The classic gradient model assumes a smooth medium, i.e. without disorder. However, most phenomena in nature exhibit some disorder due to impurities entering the systems or to materials which have defects or inhomogeneities. In this proposal, we will mainly explore the effects of disorder on continuous gradient models which is an almost unchartered territory mathematically. I will seek to answer questions such as whether the addition of a small amount of disorder modifies the nature of the phase transitions of the underlying homogeneous gradient model, i.e. if disorder is relevant, I will aim to identify non-standard phase transitions, to find new instances of universality behaviour, and to create connections between gradients and other models with disorder by taking questions from d=1 (polymers) to the next level d>1 (gradients), e.g. quenched vs annealed free energy.
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