We consider the (1+1) dimensional Laplacian model with pinning interaction. This is a probabilistic model for a polymer or an interface that is attracted to the zero line. Without the pinning interaction, the Laplacian model is a Gaussian field (Φi)iEΛN, where ΛN = {1, 2, ..., N - 1}. The covariance matrix of this field is given by the inverse of Φ -> 1/2 ENi=0(ΔΦi)2, where Δ is the discrete Laplacian. Furthermore the values at {-1, 0, N, N+1} are fixed boundary values. The pinning interaction is introduced by giving the field a reward each time it touches the zero line. Depending on the reward the model with pinning and the one without pinning show different behaviour. Caravenna and Deuschel [10] study the localisation behaviour of the polymer. The model is delocalised if the number of times a typical field touches the zero line is of order o(N). The authors of [10] show that for zero boundary conditions there is a critical reward such that for smaller rewards the model is delocalised whilst for larger rewards the model is localised. In this thesis we study the behaviour of the empirical profile of the field. We show that for non zero boundary conditions there is a critical reward such that for smaller rewards the empirical profile for the model with pinning and the one for the model without pinning behave in the same way whilst for larger rewards the empirical profile of the model with pinning interaction is attracted to the zero line.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:687152 |
| Date | January 2015 |
| Creators | Kister, Alexander Karl |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://wrap.warwick.ac.uk/79699/ |
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