Asymptotic Properties of Random Unlabelled Block-Weighted Graphs

We study the asymptotic shape of random unlabelled graphs subject to certain subcriticality conditions. The graphs are sampled with probability proportional to a product of Boltzmann weights assigned to their $2$-connected components. As their number of vertices tends to infinity, we show that they admit the Brownian tree as Gromov–Hausdorff–Prokhorov scaling limit, and converge in a strengthened Benjamini–Schramm sense toward an infinite random graph. We also consider models of random graphs that are allowed to be disconnected. Here a giant connected component emerges and the small fragments converge without any rescaling towards a finite random limit graph. Our main application of these general results treats subcritical classes of unlabelled graphs. We study the special case of unlabelled outerplanar graphs in depth and calculate its scaling constant.

On spectral asymptotic of quasi-exactly solvable quartic potential

Motivated by the earlier results of Masoero and De Benedetti (Nonlinearity 23:2501, 2010) and Shapiro et al. (Commun Math Phys 311(2):277–300, 2012), we discuss below the asymptotic of the solvable part of the spectrum for the quasi-exactly solvable quartic oscillator. In particular, we formulate a conjecture on the coincidence of the asymptotic shape of the level crossings of the latter oscillator with the asymptotic shape of zeros of the Yablonskii–Vorob’ev polynomials. Further we present a numerical study of the spectral monodromy for the oscillator in question.

Origin of the Type III Functional Response

In this chapter I review the many ways that functional responses may show a sigmoidal shape rather than the simpler asymptotic shape. I break down the potential for prey dependence of the space clearance rate through effects on each of the component mechanisms. Given the emergent nature of the functional response, type III curves can arise through density dependence of the probability of successful capture, prey detectability, and predator–prey encounter rates. Given the variety of mechanisms, it may be possible that there are really multiple types of type III curve. I also raise some concerns with the standard type III model and offer an alternative model that gets around these problems.

The asymptotic shape theorem for the frog model on finitely generated abelian groups

We study the frog model on Cayley graphs of groups with polynomial growth rate $D \geq 3$. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape.

Corrigendum to: Asymptotic shape and the speed of propagation of continuous-time continuous-space birth processes

We would like to correct the statement of Lemma 4.1 in [BDK+18].

Log-log linearity of the asymptotic distribution - a valid indicator of multi-fractality?

<p>The asymptotic shape of the marginal frequency distribution of geochemical variables has been proposed as indicator of multi-fractality. Transition into a certain statistical regime as inferred from the distribution function may be considered as criterion to delineate geochemical anomalies, including mineral resources or pollutants such as radioactive fallout or geogenic radon.</p><p>The argument is that asymptotic linearity in log-log scale, log(F(z)) = a - b log(z) as z→∞, b>0 a constant, indicates multi-fractality.</p><p>We discuss this with respect to two issues:</p><p>(1) What are the consequences of estimating the slope b for non-ergodic, in particular non-representative and preferential sampling schemes, as often the case in geochemical or pollution surveys?</p><p>(2) Frequently in geochemistry, multiplicative cascades are considered valid generators of multifractal fields, corroborated by observed f(α) functions and variograms (Matèrn or power, for low lags). This generator leads to marginally asymptotically (high cascade orders) log-normal distributions, which in log-log scale are asymptotically (high z) parabolic, not linear.</p><p>Theoretical aspects are addressed as well as examples given.</p>

Random Enriched Trees with Applications to Random Graphs

We establish limit theorems that describe the asymptotic local and global geometric behaviour of random enriched trees considered up to symmetry. We apply these general results to random unlabelled weighted rooted graphs and uniform random unlabelled $k$-trees that are rooted at a $k$-clique of distinguishable vertices. For both models we establish a Gromov–Hausdorff scaling limit, a Benjamini–Schramm limit, and a local weak limit that describes the asymptotic shape near the fixed root.