Can we give a theoretical description of why we get power laws and scaling, beyond analogies to equilibrium critical phenomena? We found a mean-field theory for the problem, valid above dimension d=6. Then Karin Dahmen and I produced a expansion for the critical exponents as a function of epsilon = 6-d. This problem turns out to be simpler in many ways than the expansion for depinning transitions we based our methods on, so our results were correspondingly more complete: Dahmen has been able to both (a) calculate the avalanche critical exponents tau and sigma explicitly, to O(epsilon), and (b) map the expansion for the traditional exponents beta, delta, and nu onto the calculation for the pure Ising model in two lower dimensions. Olga Perkovi`c and Karin and I have been running extensive numerical simulations to test these exponents in dimensions d=3, 4, and 5. The agreement is excellent.
Numerical values of the universal critical exponents, compared to the predictions of the epsilon-expansion. The numerical values were derived from hundreds of runs of sizes 320^3, 80^4, and 30^5; the dashed lines are predictions of our epsilon-expansion to linear order in epsilon = 6-d. The dotted line is our Borel re-summation of the epsilon expansion for 1/nu, which is known (from the mapping to the pure Ising model) to epsilon^5.
James P. Sethna, email@example.com
Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).