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In this note we continue the study of imaginary multiplicative chaos μβ:=exp(iβΓ), where Γ is a two-dimensional continuum Gaussian free field. We concentrate here on the fine-scale analytic properties of |μβ(Q(x,r))| as r→0, where Q(x,r) is a square of side-length 2r centred at x. More precisely, we prove monofractality of this process, a law of the iterated logarithm as r→0 and analyse its exceptional points, which have a close connection to fast points of Brownian motion. Some of the technical ideas developed to address these questions also help us pin down the exact Besov regularity of imaginary chaos, a question left open in [JSW20]. All the mentioned properties illustrate the noise-like behaviour of the imaginary chaos. We conclude by proving that the processes x↦|μβ(Q(x,r))|2, when normalised additively and multiplicatively, converge as r→0 in law, but not in probability, to white noise; this suggests that all the information of the multiplicative chaos is contained in the angular parts of μβ(Q(x,r)).

In this article we study decreasing and increasing factorisations of the cycle, which are decompositions of the cycle (12…n) into a product of n−1 transpositions satisfying monotonicity conditions. We establish a bijection between such factorisations and plane trees with n vertices. This will allow us to study some of their combinatorial properties, as well as a geometric representation in terms of laminations, which are non-crossing line segments in the unit disk. One of our main results is a limit theorem for the lamination process encoded by a decreasing factorisation of size n chosen uniformly at random.

The paper has two objectives: proving that the rate of convergence in distribution for mean-field models with interaction strength of order N−1∕2 is N−1∕2, and obtaining explicit expressions for the infinitesimal generators of two types of measure-valued Markov processes (conditional law of McKean-Vlasov processes, and empirical measures of McKean-Vlasov systems). The proof of the convergence of mean-field systems requires the second result about the generators, and both results need to study a notion of differentiability of measure-variable functions known as linear differentiability. Due to the particular framework that is studied, many technical difficulties arise compared to the existing literature. Two of the main problems are the following ones: the scaling N−1∕2 implies that the limit measure-valued processes are not deterministic, and the empirical measure processes related to McKean-Vlasov equations with jumps are necessarily discontinuous. Both properties make the expressions of the generators more complicated than what is usually considered.

For the Erdős-Rényi graph of size N with mean degree (1+o(1)) logN t+1≤d≤(1−o(1)) logN t where t∈N∗, with high probability the smallest non zero eigenvalue of the Laplacian is equal to 2−2cos(π(2t+1)−1)+o(1). This eigenvalue arises from a small subgraph isomorphic to a line of size t linked to the giant connected component by only one edge.

We consider a system of N interacting particles, described by SDEs driven by Poisson random measures, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position. When this happens, all the other particles of the system receive a small random kick which is distributed according to a heavy-tailed random variable belonging to the domain of attraction of an α− stable law and scaled by N−1∕α, where 0<α<2. We call these jumps collateral jumps. Moreover, in case 0<α<1, the jumping particle itself undergoes a macroscopic, main jump. Such systems appear in the modeling of large neural networks, such as the human brain. The particular scaling of the collateral jumps implies that the limit of the empirical measures of the system is random and equals the conditional distribution of one typical particle in the limit system, given the source of common noise. Thus the system exhibits the conditional propagation of chaos property. The limit system turns out to be solution of a non-linear SDE, driven by an α−stable process. We prove strong unique existence of the limit system and introduce a suitable coupling to obtain the strong convergence of the finite to the limit system, together with precise error bounds for finite time marginals.