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Register a new userMacroscopic control parameter for avalanche models for bursty transport
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Similarity analysis is used to identify the control parameter $R_A$ for the subset of avalanching systems that can exhibit Self- Organized Criticality (SOC). This parameter expresses the ratio of driving to dissipation. The transition to SOC, when the number of excited degrees of freedom is maximal, is found to occur when $R_A to 0$. This is in the opposite sense to (Kolmogorov) turbulence, thus identifying a deep distinction between turbulence and SOC and suggesting an observable property that could distinguish them. A corollary of this similarity analysis is that SOC phenomenology, that is, power law scaling of avalanches, can persist for finite $R_A$, with the same $R_A to 0$ exponent, if the system supports a sufficiently large range of lengthscales; necessary for SOC to be a candidate for physical ($R_A$ finite) systems.
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This article is mostly based on a talk I gave at the March 2021 meeting (virtual) of the American Physical Society on the occasion of receiving the Dannie Heineman prize for Mathematical Physics from the American Institute of Physics and the American Physical Society. I am greatly indebted to many colleagues for the results leading to this award. To name them all would take up all the space allotted to this article. (I have had more than 200 collaborators so far), I will therefore mention just a few: Michael Aizenman, Bernard Derrida, Shelly Goldstein, Elliott Lieb, Oliver Penrose, Errico Presutti, Gene Speer and Herbert Spohn. I am grateful to all of my collaborators, listed and unlisted. I would also like to acknowledge here long time support form the AFOSR and the NSF.
In planar lattice statistical mechanics models like coupled Ising with quartic interactions, vertex and dimer models, the exponents depend on all the Hamiltonian details. This corresponds, in the Renormalization Group language, to a line of fixed points. A form of universality is expected to hold, implying that all the exponents can be expressed by exact Kadanoff relations in terms of a single one of them. This conjecture has been recently established and we review here the key step of the proof, obtained by rigorous Renormalization Group methods and valid irrespectively on the solvability of the model. The exponents are expressed by convergent series in the coupling and, thanks to a set of cancellations due to emerging chiral symmetries, the extended scaling relations are proven to be true.
We review some recent developments in the study of Gibbs and non-Gibbs properties of transformed n-vector lattice and mean-field models under various transformations. Also, some new results for the loss and recovery of the Gibbs property of planar rotor models during stochastic time evolution are presented.
We obtain long series expansions for the bulk, surface and corner free energies for several two-dimensional statistical models, by combining Entings finite lattice method (FLM) with exact transfer matrix enumerations. The models encompass all integrable curves of the Q-state Potts model on the square and triangular lattices, including the antiferromagnetic transition curves and the Ising model (Q=2) at temperature T, as well as a fully-packed O(n) type loop model on the square lattice. The expansions are around the trivial fixed points at infinite Q, n or 1/T. By using a carefully chosen expansion parameter, q << 1, all expansions turn out to be of the form prod_{k=1}^infty (1-q^k)^{alpha_k + k beta_k}, where the coefficients alpha_k and beta_k are periodic functions of k. Thanks to this periodicity property we can conjecture the form of the expansions to all orders (except in a few cases where the periodicity is too large). These expressions are then valid for all 0 <= q < 1. We analyse in detail the q to 1^- limit in which the models become critical. In this limit the divergence of the corner free energy defines a universal term which can be compared with the conformal field theory (CFT) predictions of Cardy and Peschel. This allows us to deduce the asymptotic expressions for the correlation length in several cases. Finally we work out the FLM formulae for the case where some of the systems boundaries are endowed with particular (non-free) boundary conditions. We apply this in particular to the square-lattice Potts model with Jacobsen-Saleur boundary conditions, conjecturing the expansions of the surface and corner free energies to arbitrary order for any integer value of the boundary interaction parameter r. These results are in turn compared with CFT predictions.
Monodromy matrices of the $tau_2$ model are known to satisfy a Yang--Baxter equation with a six-vertex $R$-matrix as the intertwiner. The commutation relations of the elements of the monodromy matrices are completely determined by this $R$-matrix. We show the reason why in the superintegrable case the eigenspace is degenerate, but not in the general case. We then show that the eigenspaces of special CSOS models descending from the chiral Potts model are also degenerate. The existence of an $L({mathfrak{sl}}_2)$ quantum loop algebra (or subalgebra) in these models is established by showing that the Serre relations hold for the generators. The highest weight polynomial (or the Drinfeld polynomial) of the representation is obtained by using the method of Baxter for the superintegrable case. As a byproduct, the eigenvalues of all such CSOS models are given explicitly.
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