Joint distribution of the process and its sojourn time in a half-line [a,+∞)[a,+\infty) for pseudo-processes driven by a high-order heat-type equation.

Let (X(t))t≥0 be the pseudo-process driven by the high-order heat-type equation ∂u = ± ∂Nu , ∂t ∂xN where N is an integer greater than 2. We consider the sojourn time spent by (X(t))t≥0 in [a,+∞) (a ∈ R), up to a fixed time t > 0: Ta(t) = 0t 1l[a,+∞)(X(s)) ds. The purpose of this paper is to explicit the joint pseudo-distribution of the vector (Ta(t),X(t)) when the pseudo-process starts at a point x ∈ R at time 0. The method consists in solving a boundary value problem satisfied by the Laplace transform of the aforementioned distribution

On the correlation between critical points and critical values for random spherical harmonics

We study the correlation between the total number of critical points of random spherical harmonics and the number of critical points with value in any interval I ⊂ ℝ. We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random L2-norm on the sphere of the eigenfunctions, is asymptotically one. Our findings complement the results obtained by Wigman (2012) and Marinucci and Rossi (2021) on the correlation between nodal and boundary length of random spherical harmonics

An experimental study of the behaviour of two rockfills accounting for the effects of degree of saturation

Rockfill dams have become more and more recognized for their safety, economy and adaptability to widely varying site conditions. As a contribution to the understanding of the main factors affecting the rockfill behaviour, the paper reports and discusses experimental data on several aspects relevant to the interpretation and analysis of their in-situ response. The experimental programme involved three series of oedometric tests on specimens of two different gravels having the same grading, reconstituted at the same initial relative density. Experimental observations on rockfills compressibility are presented and discussed: attention is paid to the role of degree of saturation (Sr) through the analysis of "driest", "fully saturated"conditions, and the transition from one to the other. Grain crushing tests on dry and saturated soil particles are also reported. Grain size distributions of the specimens, both after compaction and after the oedometer tests, are compared in the paper. The results show that the effect of Sr cannot be overlooked in the mechanical characterization of the material, especially in rockfill/stress conditions prone to crushin

Surface tension fluctuations and a new spinodal point in glass-forming liquids

The dramatic slowdown of glass-forming liquids has been variously linked to increasing dynamic and static correlation lengths. Yet, empirical evidence is insufficient to decide among competing theories. The random first order theory (RFOT) links the dynamic slowdown to the growth of amorphous static order, whose range depends on a balance between configurational entropy and surface tension. This last quantity is expected to vanish when the temperature surpasses a spinodal point beyond which there are no metastable states. Here we measure for the first time the surface tension in a model glass-former, and find that it vanishes at the energy separating minima from saddles, demonstrating the existence of a spinodal point for amorphous metastable order. Moreover, the fluctuations of surface tension become smaller for lower temperatures, in quantitative agreement with recent theoretical speculation that spatial correlations in glassy systems relax nonexponentially because of the narrowing of the surface tension distribution.Comment: 6 pages, 5 figure

A phase-separation perspective on dynamic heterogeneities in glass-forming liquids

We study dynamic heterogeneities in a model glass-former whose overlap with a reference configuration is constrained to a fixed value. The system phase-separates into regions of small and large overlap, so that dynamical correlations remain strong even for asymptotic times. We calculate an appropriate thermodynamic potential and find evidence of a Maxwell's construction consistent with a spinodal decomposition of two phases. Our results suggest that dynamic heterogeneities are the expression of an ephemeral phase-separating regime ruled by a finite surface tension

Travelling Randomly on the Poincar\'e Half-Plane with a Pythagorean Compass

A random motion on the Poincar\'e half-plane is studied. A particle runs on the geodesic lines changing direction at Poisson-paced times. The hyperbolic distance is analyzed, also in the case where returns to the starting point are admitted. The main results concern the mean hyperbolic distance (and also the conditional mean distance) in all versions of the motion envisaged. Also an analogous motion on orthogonal circles of the sphere is examined and the evolution of the mean distance from the starting point is investigated

Cluster expansion for abstract polymer models. New bounds from an old approach

We revisit the classical approach to cluster expansions, based on tree graphs, and establish a new convergence condition that improves those by Kotecky-Preiss and Dobrushin, as we show in some examples. The two ingredients of our approach are: (i) a careful consideration of the Penrose identity for truncated functions, and (ii) the use of iterated transformations to bound tree-graph expansions.Comment: 16 pages. This new version, written en reponse to the suggestions of the referees, includes more detailed introductory sections, a proof of the generalized Penrose identity and some additional results that follow from our treatmen