## Multiplicative chaos: A simple and complete treatment of the partition function. (Chaos multiplicatif: Un traitement simple et complet de la fonction de partition.)(French)Zbl 0834.60101

Azéma, J. (ed.) et al., Séminaire de probabilités XXIX. Berlin: Springer-Verlag. Lect. Notes Math. 1613, 194-201 (1995).
Let $$X$$ be a real random variable, having exponential moments. Denote by $$\exp_0 \gamma$$ its Laplace transform, by $$I$$ the Legendre transform of $$\gamma$$, and set $$g (\beta) = \beta \gamma' (\beta) - \gamma (\beta)$$. Let $$\{X_b, b \in {\mathcal A}\}$$ be a sequence of independent random variables, having the law of $$X$$, and indexed by the finite branches of a $$d$$-tree $${\mathcal A}$$. Denote by $$S_b$$ the sum of $$X_{b'}$$ for $$b'$$ included in $$b$$, and by $$S^*_n$$ the maximum of $$S_b$$ for $$b$$ having length $$n$$. Denote also by $$\sigma_n (\beta)$$ the sum of $$\exp (\beta S_b)$$ over the $$b$$’s of length $$n$$, and set $$Z_n (\beta) = n^{-1} \text{Log} (\sigma_n (\beta))$$ and $$\beta_c = g^{-1} (\text{Log} d)$$. $$1/ \beta$$ is the temperature, $$\sigma_n$$ is the partition function, $$\lim_nZ_n$$ is the pressure, and $$1/ \beta_c$$ is the critical temperature, at which phase transition occurs. We have the theorems: 1) $$\beta_c$$ is finite if and only if $$\mathbb{P} (X = \text{ess sup} X) < d^{-1}$$; and $$\gamma' (\beta_c) = I^{-1} (\text{Log} d)$$. 2) $$Z_n (\beta)$$ goes to $$Z_\infty (\beta)$$ a.s. and in each $$L^p$$ as $$n \to \infty$$, where $$Z_\infty (\beta) = \text{Log} d + \gamma (\beta)$$ under $$\beta_c$$ and $$= \beta \gamma' (\beta_c)$$ over $$\beta_c$$. 3) $$S^*_n/n$$ goes to $$\gamma' (\beta_c)$$ a.s. and in each $$L^p$$. The proofs are intended to be elementary and short, and rely on an argument of Kahane.
For the entire collection see [Zbl 0826.00027].
Reviewer: J.Franchi (Paris)

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
Full Text: