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The authors of [Ann. Henri Poincaré 16 (2015) 691-708] introduced a multi-species version of the Sherrington-Kirkpatrick model and suggested the analogue of the Parisi formula for the free energy. Using a variant of Guerra's replica symmetry breaking interpolation, they showed that, under certain assumption on the interactions, the formula gives an upper bound on the limit of the free energy. In this paper we prove that the bound is sharp. This is achieved by developing a new multi-species form of the Ghirlanda-Guerra identities and showing that they force the overlaps within species to be completely determined by the overlaps of the whole system.

This paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix. The analysis requires a matrix extension of the scalar concentration theory developed by Sourav Chatterjee using Stein's method of exchangeable pairs. When applied to a sum of independent random matrices, this approach yields matrix generalizations of the classical inequalities due to Hoeffding, Bernstein, Khintchine and Rosenthal. The same technique delivers bounds for sums of dependent random matrices and more general matrix-valued functions of dependent random variables.

We first prove some general results on pathwise uniqueness, comparison property and existence of nonnegative strong solutions of stochastic equations driven by white noises and Poisson random measures. The results are then used to prove the strong existence of two classes of stochastic flows associated with coalescents with multiple collisions, that is, generalized Fleming— Viot flows and flows of continuous-state branching processes with immigration. One of them unifies the different treatments of three kinds of flows in Bertoin and Le Gall [Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 307— 333]. Two scaling limit theorems for the generalized Fleming—Viot flows are proved, which lead to sub-critical branching immigration superprocesses. From those theorems we derive easily a generalization of the limit theorem for finite point motions of the flows in Bertoin and Le Gall [Illinois J. Math. 50 (2006) 147—181].

Let (W, W′) be an exchangeable pair. Assume that E(W − W′|W) = g(W) + r(W), where g(W) is a dominated term and r(W) is negligible. Let $G(t)=\\int_{0}^{t}g(s)ds$ and define $p(t)=c_{1}e^{-c_{0}G(t)}$ , where c₀ is a properly chosen constant and $c_{1}=1/\\int_{-\\infty}^{\\infty }e^{-c_{0}G(t)}dt$ . Let Y be a random variable with the probability density function p. It is proved that W converges to Y in distribution when the conditional second moment of (W − W′) given W satisfies a law of large numbers. A Berry—Esseen type bound is also given. We use this technique to obtain a Berry—Esseen error bound of order 1/√n in the noncentral limit theorem for the magnetization in the Curie—Weiss ferromagnet at the critical temperature. Exponential approximation with application to the spectrum of the Bernoulli—Laplace Markov chain is also discussed.

Let $L$ be a countable language. We say that a countable infinite $L$ -structure ${\\mathcal{M}}$ admits an invariant measure when there is a probability measure on the space of $L$ -structures with the same underlying set as ${\\mathcal{M}}$ that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of ${\\mathcal{M}}$ . We show that ${\\mathcal{M}}$ admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in $\\text{Aut}({\\mathcal{M}})$ of an arbitrary finite tuple of ${\\mathcal{M}}$ fixes no additional points. When ${\\mathcal{M}}$ is a Fraïssé limit in a relational language, this amounts to requiring that the age of ${\\mathcal{M}}$ have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

Λ-coalescents model the evolution of a coalescing system in which any number of components randomly sampled from the whole may merge into larger blocks. This survey focuses on related combinatorial constructions and the large-sample behaviour of the functionals which characterize in some way the speed of coalescence.

We study sequences of noncommutative random variables which are invariant under \"quantum transformations\" coming from an orthogonal quantum group satisfying the \"easiness\" condition axiomatized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups S n , O n , which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of Köstler and Speicher, and the characterization of operator-valued free semicircular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman.

We generalize Lindeberg's proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions of exchangeable random variables. This theorem allows us to identify, for the first time, the limiting spectral distributions of Wigner matrices with exchangeable entries.

For positive q ≠ 1, the q-exchangeability of an infinite random word is introduced as quasi-invariance under permutations of letters, with a special cocycle which accounts for inversions in the word. This framework allows us to extend the q-analog of de Finetti's theorem for binary sequences—see Greschonig and Schmidt [Colloq. Math. 84/85 (2000) 495–514]—to general real-valued sequences. In contrast to the classical case of exchangeability (q = 1), the order on ${\\Bbb R}$ plays a significant role for the q-analogs. An explicit construction of ergodic q-exchangeable measures involves random shuffling of ${\\Bbb N}$ = {1, 2,...} by iteration of the geometric choice. Connections are established with transient Markov chains on q-Pascal pyramids and invariant random flags over the Galois fields.

We consider the random walk Metropolis algorithm on ℝn with Gaussian proposals, and when the target probability measure is the n-fold product of a one-dimensional law. In the limit n → ∞, it is well known (see [Ann. Appl. Probab. 7 (1997) 110–120]) that, when the variance of the proposal scales inversely proportional to the dimension n whereas time is accelerated by the factor n, a diffusive limit is obtained for each component of the Markov chain if this chain starts at equilibrium. This paper extends this result when the initial distribution is not the target probability measure. Remarking that the interaction between the components of the chain due to the common acceptance/rejection of the proposed moves is of mean-field type, we obtain a propagation of chaos result under the same scaling as in the stationary case. This proves that, in terms of the dimension n, the same scaling holds for the transient phase of the Metropolis–Hastings algorithm as near stationarity. The diffusive and mean-field limit of each component is a diffusion process nonlinear in the sense of McKean. This opens the route to new investigations of the optimal choice for the variance of the proposal distribution in order to accelerate convergence to equilibrium (see [Optimal scaling for the transient phase of Metropolis–Hastings algorithms: The longtime behavior Bernoulli (2014) To appear]).