Week 21.10.2024 – 27.10.2024

We provide an "optimal fluctuation" approach which allows us to examine the statistics of stretched 2D fractal polymer chains near an impermeable disc. We find that the span of the polymer away from the surface scales with the Kardar-Parisi-Zhang (KPZ) growth exponent 1/3, for any fractal dimension of the polymer. We pay attention to the mathematical analogy of the model under consideration with 1D Balagurov-Vaks trapping problem related to 1D Anderson localization. In parallel we consider statistics of nonuniform 1D random walks  as a "mean-field" approximation of Edelman-Dimitriu approach to RMT and derive the KPZ scaling for the mean-square random walk displacement. We discuss possible applications of obtained results for transport properties in laminar flows of liquids in corrugated channels.

We will discuss recent results obtained with Amine Asselah and partly with Perla Sousi, concerning the intersection of tree-like random graphs, which includes critical percolation clusters and critical branching random walks ranges in high dimension. One important ingredient of the proofs is a new bound on the n-th point function in percolation and the moments of local times for branching random walks, which may be of independent interest. 

A slope p/q is characterising for a knot K if the oriented homeomorphism type of the 3-manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. For any knot K, there exists a bound C(K) such that any slope p/q with |q|≥C(K) is characterising for K. This bound has previously been constructed for certain classes of knots, including torus knots, hyperbolic knots and composite knots. In this talk, I will give an overview of joint work with Patricia Sorya in which we complete this realisation problem for all remaining knots.

For a multi-agent system to respond effectively to evolving environmental conditions, proper information exchange among its units is paramount. This information transfer can either take the form of a simple contagion—stemming from pairwise interactions—or a complex contagion—involving social influence and reinforcement.
It is worth noting that the concept of complex contagion has so far been limited to a specific class of collective decision-making process, namely nonlinear binary-option models with a threshold. The interest in threshold models can be traced to their mathematical simplicity, their paradigmatic nature, and their success in modeling the spread of behaviors in various social settings. However, many collective decision-making processes encountered in social and biological systems are devoid of any threshold or nonlinearities, and involve continuous decision variables.
In this talk, we will discuss the generalization of the concept of complex contagion to consensus-based decision-making processes. Specifically, we will present some recent results revealing that a transition from simple to complex contagion, as originally identified in threshold-based models, can also be exhibited by another general class of consensus-based decision-making processes. Using concepts borrowed from network science, we identified a new way of characterizing complex contagions, and used it to uncover their existence in consensus-based dynamics.
The results discussed in this talk have far-reaching implications. First, they extend the concept of transition from simple to complex contagion—heretofore limited to binary threshold-based models—to the continuous class of consensus-based models. Second, these results reveal that the nature of the contagion—simple or complex—is directly related to the type of behavior spreading, and specifically to the pace of its intrinsic dynamics—e.g., slow external perturbations vs. collective startle response.