arXiv subjects

Peter N. Meisinger

Publications and source records attributed to Peter N. Meisinger.

42 records · Page 3Linked to original sources

arXiv1995

SPHERICALLY SYMMETRIC RANDOM WALKS I. REPRESENTATION IN TERMS OF ORTHOGONAL POLYNOMIALS

Spherically symmetric random walks in arbitrary dimension $D$ can be described in terms of Gegenbauer (ultraspherical) polynomials. For example, Legendre polynomials can be used to represent the special case of two-dimensional spherically symmetric random walks. In general, there is a connection between orthogonal polynomials and semibounded one-dimensional random walks; such a random walk can be viewed as taking place on the set of integers $n$, $n=0,~1,~2,~\ldots$, that index the polynomials. This connection allows one to express random-walk probabilities as weighted inner products of the polynomials. The correspondence between polynomials and random walks is exploited here to construct and analyze spherically symmetric random walks in $D$-dimensional space, where $D$ is {\sl not} restricted to be an integer. The weighted inner-product representation is used to calculate exact closed-form spatial and temporal moments of the probability distribution associated with the random walk. The polynomial representation of spherically symmetric random walks is also used to calculate the two-point Green's function for a rotationally symmetric free scalar quantum field theory.

hep-lat↗

arXiv1995

SPHERICALLY SYMMETRIC RANDOM WALKS II. DIMENSIONALLY DEPENDENT CRITICAL BEHAVIOR

A recently developed model of random walks on a $D$-dimensional hyperspherical lattice, where $D$ is {\sl not} restricted to integer values, is extended to include the possibility of creating and annihilating random walkers. Steady-state distributions of random walkers are obtained for all dimensions $D>0$ by solving a discrete eigenvalue problem. These distributions exhibit dimensionally dependent critical behavior as a function of the birth rate. This remarkably simple model exhibits a second-order phase transition with a nontrivial critical exponent for all dimensions $D>0$.

hep-lat↗

arXiv1995

SPHERICALLY SYMMETRIC RANDOM WALKS III. POLYMER ADSORPTION AT A HYPERSPHERICAL BOUNDARY

A recently developed model of random walks on a $D$-dimensional hyperspherical lattice, where $D$ is {\sl not} restricted to integer values, is used to study polymer growth near a $D$-dimensional attractive hyperspherical boundary. The model determines the fraction $P(κ)$ of the polymer adsorbed on this boundary as a function of the attractive potential $κ$ for all values of $D$. The adsorption fraction $P(κ)$ exhibits a second-order phase transition with a nontrivial scaling coefficient for $0 4$. At $D=4$ there is a tricritical point with logarithmic scaling. This model reproduces earlier results for $D=1$ and $D=2$, where $P(κ)$ scales linearly and exponentially, respectively. A crossover transition that depends on the radius of the adsorbing boundary is found.

hep-lat↗

arXiv1995

Universality in Random Walk Models with Birth and Death

Models of random walks are considered in which walkers are born at one location and die at all other locations with uniform death rate. Steady-state distributions of random walkers exhibit dimensionally dependent critical behavior as a function of the birth rate. Exact analytical results for a hyperspherical lattice yield a second-order phase transition with a nontrivial critical exponent for all positive dimensions $D\neq 2,~4$. Numerical studies of hypercubic and fractal lattices indicate that these exact results are universal. Implications for the adsorption transition of polymers at curved interfaces are discussed.

hep-lat↗

arXiv1995

An Effective Action for Finite Temperature Lattice Gauge Theories with Dynamical Fermions

Dynamical fermions induce via the fermion determinant a gauge-invariant effective action. In principle, this effective action can be added to the usual gauge action in simulations, reproducing the effects of closed fermion loops. Using lattice perturbation theory at finite temperature, we compute for staggered fermions the one-loop fermionic corrections to the spatial and temporal plaquette couplings as well as the leading $Z_N$ symmetry breaking coupling. A. Hasenfratz and T. DeGrand have shown that $β_c$ for dynamical staggered fermions can be accurately estimated by the formula $β_c = β^{\rm pure}_c - Δβ_F$ where $Δβ_F$ is the shift induced by the fermions at zero temperature. Numerical and analytical results indicate that the finite temperature corrections to the zero-temperature calculation of A. Hasenfratz and T. DeGrand are small for small values of $κ= {1\over 2m_F}$, but become significant for intermediate values of $κ$. The effect of these finite temperature corrections is to ruin the agreement of the Hasenfratz-DeGrand calculation with Monte Carlo data. We argue, however, that the finite temperature corrections are suppressed nonperturbatively at low temperatures, resolving this apparent disagreement. The $Z_N$ symmetry breaking coupling is small; we argue that it changes the order of the transition while having little effect on the critical value of $β$.

hep-lat↗

arXiv1994

An Effective Action for Finite Temperature QCD with Fermions

Using lattice perturbation theory at finite temperature, we compute for staggered fermions the one-loop fermionic corrections to the spatial and temporal plaquette couplings as well as the leading $Z_N$ symmetry breaking coupling. Numerical and analytical considerations indicate that the finite temperature corrections to the zero-temperature calculation of A. Hasenfratz and T. DeGrand are small for small values of $κ= {1\over 2m_F}$, but become significant for intermediate values of $κ$. The effect of these finite temperature corrections is to ruin the agreement of the Hasenfratz-DeGrand calculation with Monte Carlo data. We conjecture that the finite temperature corrections are suppressed nonperturbatively at low temperatures, resolving this apparent disagreement. The $Z_N$ symmetry breaking coupling is small; we argue that it may change the order of the transition while having little effect on the critical value of $β$.

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