Free cumulants were introduced as the proper analogue of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the combinatorics of classical cumulants is well expressed in terms of set partitions, that of free cumulants is described and often introduced in terms of non-crossing set partit...

In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean, and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in commutative, free and Boolean cases. The way we define (generalized) cumulants ...

We have calculated the large-q expansion for the energy cumulants and the magnetization cumulants at the phase transition point in the twodimensional q-state Potts model to the 21st or 23rd order in 1/ √ q using the finite lattice method. The obtained series allow us to give very precise estimates of the cumulants for q > 4 on the first order transition point. The result confirms us the correct...

Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting. It turns out that the essential property needed is exchangeability of random variables. Roughly speaking the formula says that cumulants are moments of a certain “discrete Fourier transform” of a random variable. This provides a simple unified method to understand ...

Spatially distributed natural phenomena represent complex non-linear and non-Gaussian systems. Currently, their spatial distributions are typically studied using second-order spatial statistical models, which are limiting considering the spatial complexity of natural phenomena such as geological applications. High-order geostatistics is a new area of research based on higher-order spatial conne...

The q-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of its moments in terms of matchings where q follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial propertie...

Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu’s theory of free probability. The relation between free moments and free cumulants is usually described in terms of Möbius calculus over the lattice of non-crossing partitions. In this work we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf al...

In this paper, we present an extension of linear communication channels identification algorithms to non linear channels using higher order cumulants (HOC). In the one hand, we develop a theoretical analysis of non linear quadratic systems using second and third order cumulants. In the other hand, the relationship linking cumulants and the coefficients of non linear channels presented in the li...

We have calculated the large-q expansion for the energy and magnetization cumulants at the first order phase transition point in the two-dimensional q-state Potts model to the 21st or 23rd order in 1/ √ q using the finite lattice method. The obtained series allow us to give highly convergent estimates of the cumulants for q > 4. The results confirm us the correctness of the conjecture by Bhatta...