On the upper critical dimension of Bernoulli percolation
We derive a set of inequalities for the d-dimensional independent percolation problem. Assuming the existence of critical exponents, these inequalities imply: \[ f+\nu \geq 1+\beta_ Q,\quad \mu +\nu \geq 1+\beta_ Q,\quad \zeta \geq \min \{1,\nu '/\nu \}, \] where the above exponents are f: the flow constant exponent, \(\nu\) (\(\nu\) '): the correlation length exponent below (above) threshold, \(\mu\) : the surface tension exponent, \(\beta_ Q:\) the backbone density exponent and \(\zeta\) : the chemical distance exponent. Note that all of these inequalities are mean-field bounds, and that they relate the exponent \(\nu\) defined from below the percolation threshold to exponents defined from above threshold. Furthermore, we combine the strategy of the proofs of these inequalities with notions of finite-size scaling to derive: \[ \max \{d\nu,d\nu '\}\geq 1+\beta_ Q, \] where d is the lattice dimension. Since \(\beta_ Q\geq 2\beta\), where \(\beta\) is the percolation density exponent, the final bound implies that, below six dimensions, the standard order parameter and correlation length exponents cannot simultaneously assume their mean- field values; hence an implicit bound on the upper critical dimension: \(d_ c\geq 6\).
scientific article; zbMATH DE number 4020168
scientific article