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We pursue the analysis made in [1] on the arithmetic area enclosed by m closed Brownian paths. We pay a particular attention to the random variable S{n1,n2, ...,n} (m) which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2, ...,nm times by path m. Various results are obtained in the asymptotic limit m->infinity. A key observation is that, since the paths are independent, one can use in the m paths case the SLE information, valid in the 1-path case, on the 0-winding sectors arithmetic area.
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We consider planar quadrangulations with three marked vertices and discuss the geometry of triangles made of three geodesic paths joining them. We also study the geometry of minimal separating loops, i.e. paths of minimal length among all closed paths passing by one of the three vertices and separating the two others in the quadrangulation. We concentrate on the universal scaling limit of large quadrangulations, also known as the Brownian map, where pairs of geodesic paths or minimal separating loops have common parts of non-zero macroscopic length. This is the phenomenon of confluence, which distinguishes the geometry of random quadrangulations from that of smooth surfaces. We characterize the universal probability distribution for the lengths of these common parts.
The principle of maximum irreversible is proved to be a consequence of a stochastic order of the paths inside the phase space; indeed, the system evolves on the greatest path in the stochastic order. The result obtained is that, at the stability, the entropy generation is maximum and, this maximum value is consequence of the stochastic order of the paths in the phase space, while, conversely, the stochastic order of the paths in the phase space is a consequence of the maximum of the entropy generation at the stability.
We obtain in exact arithmetic the order 24 linear differential operator $L_{24}$ and right hand side $E^{(5)}$ of the inhomogeneous equation$L_{24}(Phi^{(5)}) = E^{(5)}$, where $Phi^{(5)} =tilde{chi}^{(5)}-tilde{chi}^{(3)}/2+tilde{chi}^{(1)}/120$ is a linear combination of $n$-particle contributions to the susceptibility of the square lattice Ising model. In Bostan, et al. (J. Phys. A: Math. Theor. {bf 42}, 275209 (2009)) the operator $L_{24}$ (modulo a prime) was shown to factorize into $L_{12}^{(rm left)} cdot L_{12}^{(rm right)}$; here we prove that no further factorization of the order 12 operator $L_{12}^{(rm left)}$ is possible. We use the exact ODE to obtain the behaviour of $tilde{chi}^{(5)}$ at the ferromagnetic critical point and to obtain a limited number of analytic continuations of $tilde{chi}^{(5)}$ beyond the principal disk defined by its high temperature series. Contrary to a speculation in Boukraa, et al (J. Phys. A: Math. Theor. {bf 41} 455202 (2008)), we find that $tilde{chi}^{(5)}$ is singular at $w=1/2$ on an infinite number of branches.
Given $n$ points in the plane, a emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of $n$ points in the plane admits a (possibly self-crossi ng) covering path consisting of $n/2 +O(n/log{n})$ straight line segments. If the path is required to be noncrossing, we prove that $(1-eps)n$ straight line segments suffice for a small constant $eps>0$, and we exhibit $n$-element point sets that require at least $5n/9 -O(1)$ segments in every such path. Further, the analogous question for noncrossing emph{covering trees} is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for $n$ points in the plane requires $Omega(n log{n})$ time in the worst case.
A directed path in the vicinity of a hard wall exerts pressure on the wall because of loss of entropy. The pressure at a particular point may be estimated by estimating the loss of entropy if the point is excluded from the path. In this paper we determine asymptotic expressions for the pressure on the X-axis in models of adsorbing directed paths in the first quadrant. Our models show that the pressure vanishes in the limit of long paths in the desorbed phase, but there is a non-zero pressure in the adsorbed phase. We determine asymptotic approximations of the pressure for finite length Dyck paths and directed paths, as well as for a model of adsorbing staircase polygons with both ends grafted to the X-axis.
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