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Register a new userContinuum directed random polymers on disordered hierarchical diamond lattices
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Jeremy Clark
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I discuss models for a continuum directed random polymer in a disordered environment in which the polymer lives on a fractal called the textit{diamond hierarchical lattice}, a self-similar metric space forming a network of interweaving pathways. This fractal depends on a branching parameter $bin mathbb{N}$ and a segmenting number $sin mathbb{N}$. For $s>b$ my focus is on random measures on the set of directed paths that can be formulated as a subcritical Gaussian multiplicative chaos. This path measure is analogous to the continuum directed random polymer introduced by Alberts, Khanin, Quastel [Journal of Statistical Physics textbf{154}, 305-326 (2014)].
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We construct and study a family of continuum random polymer measures $mathbf{M}_{r}$ corresponding to limiting partition function laws recently derived in a weak-coupling regime of polymer models on hierarchical graphs with marginally relevant disorder. The continuum polymers are identified with isometric embeddings of the unit interval $[0,1]$ into a compact diamond fractal with Hausdorff dimension two, and there is a natural probability measure, $mu$, identifiable as being `uniform over the space of continuum polymers, $Gamma$. Realizations of the random measures $mathbf{M}_{r}$ exhibit strong localization properties in comparison to their subcritical counterparts when the diamond fractal has dimension less than two. Whereas two directed paths $p,qin Gamma$ chosen independently according to the pure measure $mu$ have only finitely many intersections with probability one, a realization of the disordered product measure $ mathbf{M}_{r}times mathbf{M}_{r}$ a.s. assigns positive weight to the set of pairs of paths $(p,q)$ whose intersection sets are uncountable but with Hausdorff dimension zero. We give a more refined characterization of the size of these dimension zero sets using generalized (logarithmic) Hausdorff measures. The law of the random measure $mathbf{M}_{r}$ cannot be constructed as a subcritical Gaussian multiplicative chaos because the coupling strength to the Gaussian field would, in a formal sense, have to be infinite.
We study sums of directed paths on a hierarchical lattice where each bond has either a positive or negative sign with a probability $p$. Such path sums $J$ have been used to model interference effects by hopping electrons in the strongly localized regime. The advantage of hierarchical lattices is that they include path crossings, ignored by mean field approaches, while still permitting analytical treatment. Here, we perform a scaling analysis of the controversial ``sign transition using Monte Carlo sampling, and conclude that the transition exists and is second order. Furthermore, we make use of exact moment recursion relations to find that the moments $<J^n>$ always determine, uniquely, the probability distribution $P(J)$. We also derive, exactly, the moment behavior as a function of $p$ in the thermodynamic limit. Extrapolations ($nto 0$) to obtain $<ln J>$ for odd and even moments yield a new signal for the transition that coincides with Monte Carlo simulations. Analysis of high moments yield interesting ``solitonic structures that propagate as a function of $p$. Finally, we derive the exact probability distribution for path sums $J$ up to length L=64 for all sign probabilities.
We study the directed polymer model for general graphs (beyond $mathbb Z^d$) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an $L^2$ region, and of very strong disorder, in terms of properties of the graph and of the random walk. We study in some detail (biased) random walk on various trees including the Galton Watson trees, and provide a range of other examples that illustrate counter-examples to intuitive extensions of the $mathbb Z^d$/SRW result.
Diamond lattices are sequences of recursively-defined graphs that provide a network of directed pathways between two fixed root nodes, $A$ and $B$. The construction recipe for diamond graphs depends on a branching number $bin mathbb{N}$ and a segmenting number $sin mathbb{N}$, for which a larger value of the ratio $s/b$ intuitively corresponds to more opportunities for intersections between two randomly chosen paths. By attaching i.i.d. random variables to the bonds of the graphs, I construct a random Gibbs measure on the set of directed paths by assigning each path an energy given by summing the random variables along the path. For the case $b=s$, I propose a scaling regime in which the temperature grows along with the number of hierarchical layers of the graphs, and the partition function (the normalization factor of the Gibbs measure) appears to converge in law. I prove that all of the positive integer moments of the partition function converge in this limiting regime. The motivation of this work is to prove a functional limit theorem that is analogous to a previous result obtained in the $b<s$ case.
We consider two models for directed polymers in space-time independent random media (the OConnell-Yor semi-discrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic analysis of exact Fredholm determinant formulas for the Laplace transform of their partition functions. In particular, we show that for large time tau, the probability distributions for the free energy fluctuations, when rescaled by tau^{1/3}, converges to the GUE Tracy-Widom distribution. We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics. For the semi-discrete directed polymer, when the drifts of a finite number of the Brownian motions forming the quenched random media are critically tuned, the statistics are instead governed by the limiting Baik-Ben Arous-Peche distributions from spiked random matrix theory. For the continuum polymer, the boundary perturbations correspond to choosing the initial data for the stochastic heat equation from a particular class, and likewise for its logarithm -- the Kardar-Parisi-Zhang equation. The Laplace transform formula we prove can be inverted to give the one-point probability distribution of the solution to these stochastic PDEs for the class of initial data.
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