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Register a new userLERW as an example of off-critical SLEs
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Two dimensional loop erased random walk (LERW) is a random curve, whose continuum limit is known to be a Schramm-Loewner evolution (SLE) with parameter kappa=2. In this article we study ``off-critical loop erased random walks, loop erasures of random walks penalized by their number of steps. On one hand we are able to identify counterparts for some LERW observables in terms of symplectic fermions (c=-2), thus making further steps towards a field theoretic description of LERWs. On the other hand, we show that it is possible to understand the Loewner driving function of the continuum limit of off-critical LERWs, thus providing an example of application of SLE-like techniques to models near their critical point. Such a description is bound to be quite complicated because outside the critical point one has a finite correlation length and therefore no conformal invariance. However, the example here shows the question need not be intractable. We will present the results with emphasis on general features that can be expected to be true in other off-critical models.
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We discuss properties of dipolar SLE(k) under conditioning. We show that k=2, which describes continuum limits of loop erased random walks, is characterized as being the only value of k such that dipolar SLE conditioned to stop on an interval coincides with dipolar SLE on that interval. We illustrate this property by computing a new bulk passage probability for SLE(2).
We prove that there exists just one pair of complex four-dimensional Lie algebras such that a well-defined contraction among them is not equivalent to a generalized IW-contraction (or to a one-parametric subgroup degeneration in conventional algebraic terms). Over the field of real numbers, this pair of algebras is split into two pairs with the same contracted algebra. The example we constructed demonstrates that even in the dimension four generalized IW-contractions are not sufficient for realizing all possible contractions, and this is the lowest dimension in which generalized IW-contractions are not universal. Moreover, this is also the first example of nonexistence of generalized IW-contraction for the case when the contracted algebra is not characteristically nilpotent and, therefore, admits nontrivial diagonal derivations. The lower bound (equal to three) of nonnegative integer parameter exponents which are sufficient to realize all generalized IW-contractions of four-dimensional Lie algebras is also found.
We study a pair of infinite dimensional dynamical systems naturally associated with the study of minimizing/maximizing functions for the Strichartz inequalities for the Schrodinger equation. One system is of gradient type and the other one is a Hamiltonian system. For both systems, the corresponding sets of critical points, their stability, and the relation between the two are investigated. By a combination of numerical and analytical methods we argue that the Gaussian is a maximizer in a class of Strichartz inequalities for dimensions one, two and three. The argument reduces to verification of an apparently new combinatorial inequality involving binomial coefficients.
Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [9]. Natural examples arise from $(1+1)$-dimensional differential operators using the model operator $D_A$ in $L^2(mathbb{R}^2; dt dx)$ of the type $D_A = (d/dt) + A$, where $A = int^{oplus}_{mathbb{R}} dt , A(t)$, and the family of self-adjoint operators $A(t)$ in $L^2(mathbb{R}; dx)$ is explicitly given by $A(t) = - i (d/dx) + theta(t) phi(cdot)$, $t in mathbb{R}$. Here $phi: mathbb{R} to mathbb{R}$ has to be integrable on $mathbb{R}$ and $theta: mathbb{R} to mathbb{R}$ tends to zero as $t to - infty$ and to $1$ as $t to + infty$. In particular, $A(t)$ has asymptotes in the norm resolvent sense $A_- = - i (d/dx)$, $A_+ = - i (d/dx) + phi(cdot)$ as $t to mp infty$. Since $D_A$ violates the relative trace class condition introduced in [9], we now employ a new approach based on an approximation technique. The approximants do fit the framework of [9] and lead to the following results: Introducing $H_1 = {D_A}^* D_A$, $H_2 = D_A {D_A}^*$, we recall that the resolvent regularized Witten index of $D_A$, denoted by $W_r(D_A)$, is defined by $$ W_r(D_A) = lim_{lambda to 0} (- lambda) {rm tr}_{L^2(mathbb{R}^2; dtdx)}((H_1 - lambda I)^{-1} - (H_2 - lambda I)^{-1}). $$ In the concrete example at hand, we prove $$ W_r(D_A) = xi(0_+; H_2, H_1) = xi(0; A_+, A_-) = 1/(2 pi) int_{mathbb{R}} dx , phi(x). $$ Here $xi(, cdot , ; S_2, S_1)$, denotes the spectral shift operator for the pair $(S_2,S_1)$, and we employ the normalization, $xi(lambda; H_2, H_1) = 0$, $lambda < 0$.
We introduce a transfer matrix formalism for the (annealed) Ising model coupled to two-dimensional causal dynamical triangulations. Using the Krein-Rutman theory of positivity preserving operators we study several properties of the emerging transfer matrix. In particular, we determine regions in the quadrant of parameters beta, mu >0 where the infinite-volume free energy converges, yielding results on the convergence and asymptotic properties of the partition function and the Gibbs measure.
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