Abstract
Fisher's fluctuationresponse relation is one of four famous scaling formulae and is consistent with a vanishing correlationfunction anomalous dimension above the upper critical dimension dc. However, it has long been known that numerical simulations deliver a negative value for the anomalous dimension there. Here, the apparent discrepancy is attributed to a distinction between the systemlength and correlation or characteristiclength scales. On the latter scale, the anomalous dimension indeed vanishes above dc and Fisher's relation holds in its standard form. However, on the scale of the system length, the anomalous dimension is negative and Fisher's relation requires modification. Similar investigations at the upper critical dimension, where dangerous irrelevant variables become marginal, lead to an analogous pair of Fisher relations for logarithmiccorrection exponents. Implications of a similar distinction between length scales in percolation theory above dc and for the Ginzburg criterion are briefly discussed.
Original language  English 

Article number  26005 
Journal  EPL 
Volume  105 
Issue number  2 
DOIs  
Publication status  Published  13 Feb 2014 
Keywords
 condensed matter: structural
 mechanical & thermal
 statistical physics and nonlinear systems
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Ralph Kenna
 Research Centre for Fluid and Complex Systems  Professor of Theoretical Physics
Person: Teaching and Research