scholarly journals Working field theory problems with random walks

2005 ◽  
Vol 24 (1) ◽  
pp. 229-237 ◽  
Author(s):  
Kent R. Davey
Keyword(s):  
Field Theory ◽  
Random Walks

scholarly journals ON BORWEIN’S CONJECTURES FOR PLANAR UNIFORM RANDOM WALKS

2019 ◽  
Vol 107 (3) ◽  
pp. 392-411 ◽  
Author(s):  
YAJUN ZHOU
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Random Walks ◽  
Euclidean Plane ◽  
Feynman Diagrams ◽  
Quantum Field ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
Uniformly Convergent

Let $p_{n}(x)=\int _{0}^{\infty }J_{0}(xt)[J_{0}(t)]^{n}xt\,dt$ be Kluyver’s probability density for $n$-step uniform random walks in the Euclidean plane. Through connection to a similar problem in two-dimensional quantum field theory, we evaluate the third-order derivative $p_{5}^{\prime \prime \prime }(0^{+})$ in closed form, thereby giving a new proof for a conjecture of J. M. Borwein. By further analogies to Feynman diagrams in quantum field theory, we demonstrate that $p_{n}(x),0\leq x\leq 1$ admits a uniformly convergent Maclaurin expansion for all odd integers $n\geq 5$, thus settling another conjecture of Borwein.

scholarly journals Field Theory Conjecture for Loop-Erased Random Walks

2008 ◽  
Vol 133 (5) ◽  
pp. 805-812 ◽  
Author(s):  
Andrei A. Fedorenko ◽  
Pierre Le Doussal ◽  
Kay Jörg Wiese
Keyword(s):  
Field Theory ◽  
Random Walks

scholarly journals An exact mapping between loop-erased random walks and an interacting field theory with two fermions and one boson

2020 ◽  
Vol 9 (5) ◽  
Author(s):  
Assaf Shapira ◽  
Kay Joerg Wiese
Keyword(s):  
Field Theory ◽  
Random Walks ◽  
Path Integral ◽  
Lattice Model ◽  
Type Theory ◽  
Complex Field ◽  
Standard Formulation ◽  
Bosonic Field ◽  
Hypercubic Lattice ◽  
Formula Type

We give a simplified proof for the equivalence of loop-erased random walks to a lattice model containing two complex fermions, and one complex boson. This equivalence works on an arbitrary directed graph. Specifying to the dd-dimensional hypercubic lattice, at large scales this theory reduces to a scalar \phi^4ϕ4-type theory with two complex fermions, and one complex boson. While the path integral for the fermions is the Berezin integral, for the bosonic field we can either use a complex field \phi(x)\in \mathbb Cϕ(x)∈ℂ (standard formulation) or a nilpotent one satisfying \phi(x)^2 =0ϕ(x)2=0. We discuss basic properties of the latter formulation, which has distinct advantages in the lattice model.

Random walks: A pedestrian approach to polymers, critical phenomena, and field theory

1991 ◽  
Vol 59 (7) ◽  
pp. 633-645 ◽  
Author(s):  
E. P. Raposo ◽  
S. M. de Oliveira ◽  
A. M. Nemirovsky ◽  
M. D. Coutinho‐Filho
Keyword(s):  
Field Theory ◽  
Random Walks ◽  
Critical Phenomena

Simulation of random walks in field theory

1988 ◽  
Vol 14 (4) ◽  
pp. 397-410 ◽  
Author(s):  
E J Janse van Rensburg
Keyword(s):  
Field Theory ◽  
Random Walks
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