arXiv subjects

Irene Hueter

Publications and source records attributed to Irene Hueter.

3 recordsLinked to original sources

arXiv2001

Anisotropic Contact Process on Homogeneous Trees

The existence of a weak survival region is established for the anisotropic symmetric contact process on a homogeneous tree T_{2d} of degree 2d > 2: For parameter values in a certain connected region of positive Lebesgue measure, the population survives forever with positive probability but ultimately vacates every finite subset of the tree with probability one. In this phase, infection trails must converge to the geometric boundary Ωof the tree. The random subset Λof the boundary consisting of all ends of the tree in which the infection survives, called the limit set of the process, is shown to have Hausdorff dimension no larger than one half the Hausdorff dimension of the entire geometric boundary. In addition, there is strict inequality at the transition between weak and strong survival except when the contact process is isotropic. It is further shown that in all cases there is a distinguished probability measure μ, supported by Ω, such that the Hausdorff dimension of Λ\cap Ω_μ, where Ω_μ is the set of μ-generic points of Ω, converges to one half the Hausdorff dimension of Ω_μ at the phase separation points. Exact formulae for the Hausdorff dimensions of Λand Λ\cap Ω_μ are obtained. We also prove that the contact process at the transition between extinction and weak survival does not survive. The method developed shows that the contact process at the phase transition to strong survival survives weakly for d > 1.

math.PR↗

arXiv2001

Proof of the Conjecture that the Planar Self-Avoiding Walk has Root Mean Square Displacement Exponent 3/4

This paper proves the long-standing open conjecture rooted in chemical physics (Flory (1949)) that the self-avoiding walk (SAW) in the square lattice has root mean square displacement exponent ν= 3/4. This value is an instance of the formula ν=1 on Z and ν= max(1/2, 1/4 + 1/d) in Z^d for dimensions d \geq 2, which will be proved in a subsequent paper. This expression differs from the one that Flory's arguments suggested. We consider (a) the point process of self-intersections defined via certain paths of the symmetric simple random walk in Z^2 and (b) a ``weakly self-avoiding cone process'' relative to this point process when in a certain "shape". We derive results on the asymptotic expected distance of the weakly SAW with parameter β>0 from its starting point, from which a number of distance exponents are immediately collectable for the SAW as well. Our method employs the Palm distribution of the point process of self-intersection points in a cone.

math.PR↗

arXiv2001

Formula for the Mean Square Displacement Exponent of the Self-Avoiding Walk in 3, 4 and All Dimensions

This paper proves the formula ν(d) =1 for d=1 and ν(d) = max(1/4 +1/d, 1/2) for d > 1 for the root mean square displacement exponent ν(d) of the self-avoiding walk (SAW) in Z^d, and thus, resolves some major long-standing open conjectures rooted in chemical physics (Flory, 1949). The values ν(2) =3/4 and ν(4) = 1/2 coincide with those that were believed on the basis of heuristic and "numerical evidence". Perhaps surprisingly, there was no precise conjecture in dimension 3. Yet as early as in the 1980ies, Monte Carlo simulations produced a couple of confidence intervals for the exponent ν(3). This work is a follow-up to Hueter (2001), which proves the result for d=2 and lays out the fundamental building blocks for the analysis in all dimensions. We consider (a) the point process of self-intersections defined via certain paths of length n of the symmetric simple random walk in Z^d and (b) a ``weakly self-avoiding cone process'' relative to this point process in a certain "shape". The asymptotic expected distance of the process in (b) can be calculated rather precisely as n tends large and, if the point process has circular shape, can be shown to asymptotically equal (up to error terms) the one of the weakly SAW with parameter β>0. From these results, a number of distance exponents are immediately collectable for the SAW as well. Our approach invokes the Palm distribution of the point process of self-intersections in a cone.

math.PR↗