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We consider the ab initio continuum theory of polymer chains in a theta solvent, interacting via two- and three-body forces, with a short range cutoff s0 and in continuous space dimension d(2

We show that when two boundary arcs of a Liouville quantum gravity random surface are conformally welded to each other (in a boundary length-preserving way) the resulting interface is a random curve called the Schramm-Loewner evolution. We also develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under conformal weld- ing maps related to Schramm-Loewner evolution. As an application, we construct quantum length and boundary intersection measures on the Schramm-Loewner evolution curve itself. [ABSTRACT FROM AUTHOR]

Abstract: We consider random conformally invariant paths in the complex plane (SLEs). Using the Coulomb gas method in conformal field theory, we rederive the mixed multifractal exponents associated with both the harmonic measure and winding (rotation or monodromy) near such critical curves, previously obtained by quantum gravity methods. The results also extend to the general cases of harmonic measure moments and winding of multiple paths in a star configuration. [Copyright &y& Elsevier]

We give here new results of topology and integral geometry concerning the Gauss linking number I of closed manifolds in n-dimensional space. The rigid manifolds have arbitrary shapes and dimensions, and are statistically at random positions in ℝ. Generalizing Pohl's work, for two closed manifolds C, C, of respective dimensions r and s, with 0≦ r≦ n−1, and r+ s+1= n, we consider the 'kinematic linking integral' I=<∝ I( x, O) d x>, of the square linking number I of C and C, over the group of Euclidean motions of one manifold (translations x, rotations O). Introducing a new tensorial method, and using group theory, we show quite generally that I=num. fact. $$\int\limits_0^\infty {d\varrho [\mathcal{A}_1 (\varrho )\mathcal{A}_2 (\varrho ) + \delta _{r,s} \mathcal{B}_1 (\varrho )\mathcal{B}_2 (\varrho )]} $$ , where ϱ is a length variable and where A,ℬ ℬ(α=1, 2) are characteristic functions associated with the manifold C only. We study functions A and ℬ of a manifold C, of dimension r, in all cases 0≦ r≦ n−1. A always exists. A(0) gives C's area, whereas $$\int\limits_0^\infty {\mathcal{A}(\varrho )d\varrho } $$ equals the interior volume of a hypersurface C. ℬ is found to exist and not to vanish only if 2 dim C+1= n and n=3+4 q=3, 7, 11 ... A and ℬ are explicitly calculated for segments and r-spheres S. As an application the topological excluded volume of a gas of nonlinked spheres S moving in ℝ is calculated. We generalize to N manifolds C, α=1, ..., N, linked successively to each other and forming a ring. The cyclic product of their linking numbers is integrated over the group of motions of the manifolds. It is shown to factorize completely in Fourier space, with special algebraic rules, over the set of 2 N characteristic functions A, ℬ, associated with the C's. The same algebra of characteristic functions is shown to describe a larger class of topology and electromagnetism properties: a new theorem is given for a family of Euclidean group integrals involving the random linking numbers, mutual inductances and contact distributions of N manifolds. [ABSTRACT FROM AUTHOR]

We present a (mathematically rigorous) probabilistic and geometrical proof of the Knizhnik-Polyakov-Zamolodchikov relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the properly regularized quantum area measure dmicro&#95;{gamma}=epsilon;{gamma;{2}/2}e;{gammah&#95;{epsilon}(z)}dz, where dz is the Lebesgue measure on D, gamma is a real parameter, 02 is shown to be related to the quantum measure dmu&#95;{gamma;{'}}, gamma;{'}<2, by the fundamental duality gammagamma;{'}=4.

The exact joint multifractal distribution for the scaling and winding of the electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum f(alpha,lambda) gives the Hausdorff dimension of the points where the potential scales with distance r as H approximately r(alpha) while the curve logarithmically spirals with a rotation angle phi=lambdalnr. It obeys the scaling law f(alpha,lambda)=(1+lambda(2))f(alpha)-blambda(2) with alpha=alpha/(1+lambda(2)) and b=(25-c)/12, and where f(alpha) identical with f(alpha,0) is the pure harmonic measure spectrum, and c the conformal central charge. The results apply to O(N) and Potts models, as well as to stochastic Löwner evolution.