English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This paper provides a rigorous study of the localization transition for aGaussian free field on $\mathbb{Z}^d$ interacting with a quenched disorderedsubstrate that acts on the interface when the interface height is close tozero. The substrate has the tendency to localize or repel the interface atdifferent sites and one can show that a localization-delocalization transitiontakes place when varying the average pinning potential $h$: the free energydensity is zero in the delocalized regime, that is for $h$ smaller than athreshold $h_c$, and it is positive for $h>h_c$. For $d\ge 3$ we compute $h_c$and we show that the transition happens at the same value as for the annealedmodel. However we can show that the critical behavior of the quenched modeldiffers from the one of the annealed one. While the phase transition of theannealed model is of first order, we show that the quenched free energy isbounded above by $ (h-h_c)_+^2$ times a positive constant and that, forGaussian disorder, the quadratic behavior is sharp. Therefore this provides anexample in which a {\sl relevant disorder critical exponent} can be madeexplicit: in theoretical physics disorder is said to be {\sl relevant} when thedisorder changes the critical behavior of a system and, while there are casesin which it is known that disorder is relevant, the exact critical behavior istypically unknown. For $d=2$ we are not able to decide whether the quenched andannealed critical points coincide, but we provide an upper bound for thedifference between them.

Pinning and disorder relevance for the lattice Gaussian free field