EPSRC Reference: 
EP/M027694/1 
Title: 
Continuous gradient interfaces with disorder 
Principal Investigator: 
Cotar, Professor C 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

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 ddimensional 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 twodimensional 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 highlydependent realvalued random variables whose distribution is a function of the nearestneighbour 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 ddimensional 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 nonstandard 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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