Intersection properties of simple random walks: A renormalization group approach (original) (raw)
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We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the loop-erased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in ε = 4 − d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critical dimension d = 4, and compares well with numerical studies. We obtain the universal subleading logarithmic correction in d = 4, which can be used as a further test of the conjecture.
Some exact results in branching and annihilating random walks
2012
We present some exact results on the behavior of Branching and Annihilating Random Walks, both in the Directed Percolation and Parity Conserving universality classes. Contrary to usual perturbation theory, we perform an expansion in the branching rate around the non trivial Pure Annihilation model, whose correlation and response function we compute exactly. With this, the non-universal threshold value for having a phase transition in the simplest system belonging to the Directed Percolation universality class is found to coincide with previous Non Perturbative Renormalization Group approximate results. We also show that the Parity Conserving universality class has an unexpected RG fixed point structure, with a PA fixed point which is unstable in all dimensions of physical interest.
On the critical exponent for random walk intersections
Journal of Statistical Physics, 1989
The exponent ~'a for the probability of nonintersection of two random walks starting at the same point is considered. It is proved that 1/2 < ~2 ~< 3/4. Monte Carlo simulations are done to suggest ~2 = 0.61... and ~' 3 ~ 0.29.
Branching and annihilating random walks: Exact results at low branching rate
Physical Review E, 2013
We present some exact results on the behavior of Branching and Annihilating Random Walks, both in the Directed Percolation and Parity Conserving universality classes. Contrary to usual perturbation theory, we perform an expansion in the branching rate around the non trivial Pure Annihilation model, whose correlation and response function we compute exactly. With this, the non-universal threshold value for having a phase transition in the simplest system belonging to the Directed Percolation universality class is found to coincide with previous Non Perturbative Renormalization Group approximate results. We also show that the Parity Conserving universality class has an unexpected RG fixed point structure, with a PA fixed point which is unstable in all dimensions of physical interest.
Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory
1992
Simple random walks-or equivalently, sums of independent random variables-have long been a standard topic of probability theory and mathematical physics. In the 1950's, non-Markovian random-walk models, such as the self-avoiding walk, were introduced into theoretical polymer physics, and gradually came to serve as a paradigm for the general theory of critical phenomena. In the past decade, random-walk expansions have evolved into an important tool for the rigorous analysis of critical phenomena in classical spin systems and of the continuum limit in quantum field theory. Among the results obtained by random-walk methods are the proof of triviality of the ϕ 4 quantum field theory in space-time dimension d () ≥ 4, and the proof of mean-field critical behavior for ϕ 4 and Ising models in space dimension d () ≥ 4. The principal goal of the present monograph is to present a detailed review of these developments. It is supplemented by a brief excursion to the theory of random surfaces and various applications thereof. This book has grown out of research carried out by the authors mainly from 1982 until the middle of 1985. Our original intention was to write a research paper. However, the writing of such a paper turned out to be a very slow process, partly because of our geographical separation, partly because each of us was involved in other projects that may have appeared more urgent. Meanwhile, other people and we found refinements and extensions of our original results, so that the original plan for our paper had to be revised. Moreover, a preliminary draft of our paper grew longer and longer. It became clear that our project I II Foreword-if ever completed-would not take the shape of a paper publishable in a theoretical or mathematical physics journal. We therefore decided to aim for a format that is more expository and longer than a research paper, but too specialized to represent something like a textbook on equilibrium statistical mechanics or quantum field theory. This volume reviews a circle of results in the area of critical phenomena in spin systems, lattice field theories and random-walk models which are, in their majority, due to Michael Aizenman, David Brydges, Tom Spencer and ourselves. Other people have also been involved in these collaborations, among whom one should mention
Self‐avoiding random walks: Some exactly soluble cases
Journal of Mathematical Physics, 1978
We use the exact renormalization group equations to determine the asymptotic behavior of long selfavoiding random walks on some pseudolattices. The lattices considered are the truncated 3-simplex, the truncated 4-simplex, and the modified rectangular lattices. The total number of random walks C", the number of polygons P" of perimeter n, and the mean square end to end distance (RA) are assumed to be asymptotically proportional to JL"n Y-I, JL"n a-3 , and n 2 v respectively for large n, where n is the total length of the walk. The exact values of the connectivity constant IL• and the critical exponents A, a, v are determined for the three lattices. We give an example of two lattice systems that have the same effective nonintegral dimensionality 3/2 but different values of the critical exponents y, a, and v.
In this paper we establish an exact relationship between the asymptotic probability distributions ν 0 and ν 2 of the multiple point range of the planar random walk and the proper functions Γ [0] and Γ [2] respectively of the planar, complex φ 4-theory, setting the number of components m = 0: The characteristic functions Φ 0 and Φ 2 of ν 0 and ν 2 have simple integral transforms ζ [0] and ζ [2] respectively which turn out to be the extensions of the proper functions Γ [0] and Γ [2] onto a Riemann surface (with infinitely many sheets) in the coupling constant g and are well defined mathematically. ζ [0] and ζ [2] restricted to a specific sheet have a (sector-wise) uniform asymptotic expansion in g = 0. The standard perturbation series of Γ [0] and Γ [2] in g have expansion coefficients Γ [0],pt r and Γ [2],pt r which are polynomials in m. Order by order the lowest nontrivial polynomial coefficient in m: Γ [0],pt r,1 = ζ [0] r and Γ [2],pt r,0 = ζ [2] r where ζ [0] r and ζ [2] r are the coefficients of the asymptotic series of ζ [0] and ζ [2] around g = 0 respectively. Φ 0 and Φ 2 turn out to be modified Borel type summations of those series. As an application we derive the rising edge behaviour of ν 0 and ν 2 from the large order estimates of Lipatov [15]. It turns out to be of the form of a Gamma distribution with parameters known numerically. MSC 2010 subject classifications: Primary 60J65, 60J10; secondary 60E10, 60B12. Keywords and phrases: Multiple point range of a random walk, φ 4-theory, quantum field theory, proper functions, Intersection Local Time, Range of a random walk, multiple points, Brownian motion .
Branching-annihilating random walks in one dimension: some exact results
Journal of Physics A: Mathematical and General, 1998
We derive a self-duality relation for a one-dimensional model of branching and annihilating random walkers with an even number of offsprings. With the duality relation and by deriving exact results in some limiting cases involving fast diffusion we obtain new information on the location and nature of the phase transition line between an active stationary state (non-zero density) and an absorbing state (extinction of all particles), thus clarifying some so far open problems. In these limits the transition is mean-field-like, but on the active side of the phase transition line the fluctuation in the number of particles deviates from its mean-field value. We also show that well within the active region of the phase diagram a finite system approaches the absorbing state very slowly on a time scale which diverges exponentially in system size.