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Stefan Junk

Publications and source records attributed to Stefan Junk.

18 recordsLinked to original sources
arXiv2025

Coincidence of critical points for directed polymers for general environments and random walks

For the directed polymer in a random environment (DPRE), two critical inverse-temperatures can be defined. The first one, $\beta_c$, separates the strong disorder regime (in which the normalized partition function $W^{\beta}_n$ tends to zero) from the weak disorder regime (in which $W^{\beta}_n$ converges to a nontrivial limit). The other, $\bar \beta_c$, delimits the very strong disorder regime (in which $W^{\beta}_n$ converges to zero exponentially fast). It was proved previously that $\beta_c=\bar \beta_c$ when the random environment is upper-bounded for the DPRE based on the simple random walk. We extend this result to general environment and arbitrary reference walk. We also prove that $\beta_c=0$ if and only the $L^2$-critical point is trivial.

math.PR↗
arXiv2024

Structural robustness of networks with degree-degree correlations between second-nearest neighbors

We numerically investigate the robustness of networks with degree-degree correlations between nodes separated by distance $l=2$ in terms of shortest path length. The degree-degree correlation between the $l$-th nearest neighbors can be quantified by Pearson's correlation coefficient $r_l$ for the degrees of two nodes at distance $l$. We introduce $l$-th nearest-neighbor correlated random networks ($l$-NNCRNs) that are degree-degree correlated at less than or equal to the $l$-th nearest neighbor scale and maximally random at farther scales. We generate $2$-NNCRNs with various $r_1$ and $r_2$ using two steps of random edge rewiring based on the Metropolis-Hastings algorithm and compare their robustness against failures of nodes and edges. As typical cases of homogeneous and heterogeneous degree distributions, we adopted Poisson and power law distributions. Our results show that the range of $r_2$ differs depending on the degree distribution and the value of $r_1$. Moreover, comparing $2$-NNCRNs sharing the same degree distribution and $r_1$, we demonstrate that a higher $r_2$ makes a network more robust against random node/edge failures as well as degree-based targeted attacks, regardless of whether $r_1$ is positive or negative.

physics.soc-ph↗
arXiv2024

Equivalence of fluctuations of discretized SHE and KPZ equations in the subcritical weak disorder regime

We study the fluctuations of discretized versions of the stochastic heat equation (SHE) and the Kardar-Parisi-Zhang (KPZ) equation in spatial dimensions $d\geq 3$ in the weak disorder regime. The discretization is defined using the directed polymer model. Previous research has identified the scaling limit of both equations under a suboptimal moment condition and, in particular, it was established that both converge in law to the same limit. We extend this result by showing that the fluctuations of both equations are close in probability in the subcritical weak disorder regime, indicating that they share the same scaling limit (the existence of which remains open). Our result applies under a moment condition that is expected to hold throughout the interior of the weak disorder phase, which is currently only known under a technical assumption on the environment. We also prove a lower tail concentration of the partition functions.

math.PR↗
arXiv2024

The tail distribution of the partition function for directed polymer in the weak disorder phase

We investigate the upper tail distribution of the partition function of the directed polymer in a random environment on $\mathbb Z^d$ in the weak disorder phase. We show that the distribution of the infinite volume partition function $W^{\beta}_{\infty}$ displays a power-law decay, with an exponent $p^*(\beta)\in [1+\frac{2}{d},\infty)$. We also prove that the distribution of the suprema of the point-to-point and point-to-line partition functions display the same behavior. On the way to these results, we prove a technical estimate of independent interest: the $L^p$-norm of the partition function at the time when it overshoots a high value $A$ is comparable to $A$. We use this estimate to extend the validity of many recent results that were proved under the assumption that the environment is upper bounded.

math.PR↗
arXiv2024

Strong disorder and very strong disorder are equivalent for directed polymers

We show that if the normalized partition function $W^{\beta}_n$ of the directed polymer model on $\mathbb Z^d$ converges to zero, then it does so exponentially fast. This implies that there exists a critical value $\beta_c$ for the inverse temperature such that the normalized partition function has a non-degenerate limit for all $\beta\in [0,\beta_c]$ -- weak disorder holds -- while for $\beta\in (\beta_c,\infty)$ it converges exponentially fast to zero -- very strong disorder holds. This solves a twenty-years-old conjecture formulated by Comets, Yoshida, Carmona and Hu. Our proof requires a technical assumption on the environment, namely, that it is bounded from above.

math.PR↗
arXiv2023

Local limit theorem for directed polymers beyond the $L^2$-phase

We consider the directed polymer model in the weak disorder phase under the assumption that the partition function is $L^p$-bounded for some $p>1+\frac{2}d$. We prove that the point-to-point partition function can be approximated by two point-to-plane partition functions at the startpoint and endpoint, and in particular that it is $L^p$-bounded as well. Some consequences of this result are also discussed, the most important of which is a local limit theorem for the polymer measure. We furthermore show that the required $L^p$-boundedness holds for some range of $\beta$ beyond the $L^2$-critical point, and in the whole interior of the weak disorder phase for environments with finite support.

math.PR↗
arXiv2023

Fluctuations of partition functions of directed polymers in weak disorder beyond the $L^2$-phase

We study the directed polymer model in a bounded environment in weak disorder but without $L^2$-boundedness, specifically the speed of homogenization for the field $(W_n^{0,x})_{x\in\mathbb Z^d}$, where $W_n^{0,x}$ denotes the associated martingale for the polymer starting from $x$. We show that a suitably re-centered spatial average over a set of diameter $n^{1/2}$ convergence to zero at rate $n^{-ξ+o(1)}$, where the exponent is an explicit function of the inverse temperature $β$.

math.PR↗
arXiv2023

Stability of weak disorder phase for directed polymer with applications to limit theorems

We study the directed polymer model in a bounded environment with bond disorder and show that, in the interior of the weak disorder phase, weak disorder continues to hold upon perturbation by a small bias. Using this stability result, we give a new proof for the central limit theorem (CLT) in probability for the directed polymer model in the interior of the weak disorder phase. We also show that the large deviation rate function agrees with that of the underlying random walk. For the Brownian polymer model, we improve the convergence in the CLT to almost sure convergence in the whole weak disorder phase. The main technical tools are a new moment bound from \cite{J21_1} and a quantitative comparison between the associated martingales at different inverse temperatures.

math.PR↗
arXiv2023

Moment characterization of the weak disorder phase for directed polymers in a class of unbounded environments

For a directed polymer model in random environment, a characterization of the weak disorder phase in terms of the moment of the renormalized partition function has been proved in [S. Junk: Communications in Mathematical Physics 389, 1087-1097 (2022)]. We extend this characterization to a large class of unbounded environments which includes many commonly used distributions.

math.PR↗
arXiv2022

Extremal regime for one-dimensional Mott variable-range hopping

We study the asymptotic behaviour of a version of the one-dimensional Mott random walk in a regime that exhibits severe blocking. We establish that, for any fixed time, the appropriately-rescaled Mott random walk is situated between two environment-measurable barriers, the locations of which are shown to have an extremal scaling limit. Moreover, we give an asymptotic description of the distribution of the Mott random walk between the barriers that contain it.

math.PR↗
arXiv2022

Number of paths in oriented percolation as zero temperature limit of directed polymer

We prove that the free energy of directed polymer in Bernoulli environment converges to the growth rate for the number of open paths in super-critical oriented percolation as the temperature tends to zero. Our proof is based on rate of convergence results which hold uniformly in the temperature. We also prove that the convergence rate is locally uniform in the percolation parameter inside the super-critical phase, which implies that the growth rate depends continuously on the percolation parameter.

math.PR↗
arXiv2021

New characterization of the weak disorder phase of directed polymers in bounded random environments

We show that the weak disorder phase for the directed polymer model in a bounded random environment is characterized by the integrability of the running supremum $\sup_{n\in \mathbb N}W_n^β$ of the associated martingale $(W_n^β)_{n\in \mathbb N}$. Using this characterization, we prove that $(W_n^β)_{n\in \mathbb N}$ is $L^p$-bounded in the whole weak disorder phase, for some $p>1$. The argument generalizes to non-negative martingales with a certain product structure.

math.PR↗
arXiv2020

Anomalous scaling regime for one-dimensional Mott variable-range hopping

We derive an anomalous, sub-diffusive scaling limit for a one-dimensional version of the Mott random walk. The limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to intervals of low conductance in the discrete model, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time before crossing. We also discuss how, by incorporating a Bouchaud trap model element into the setting, it is possible to combine this 'blocking' mechanism with one of 'trapping'. Our proof relies on a recently developed theory that relates the convergence of processes to that of associated resistance metric measure spaces.

math.PR↗
arXiv2020

Zero temperature limit for the Brownian directed polymer among Poissonian disasters

We study a continuum model of directed polymer in random environment. The law of the polymer is defined as the Brownian motion conditioned to survive among space-time Poissonian disasters. This model is well-studied in the positive temperature regime. However, at zero-temperature, even the existence of the free energy has not been proved. In this article, we prove that the free energy exists and is continuous at zero-temperature.

math.PR↗
arXiv2018

Comparison of partition functions in a space-time random environment

Let $Z^1$ and $Z^2$ be partition functions in the random polymer model in the same environment but driven by different underlying random walks. We give a comparison in concave stochastic order between $Z^1$ and $Z^2$ if one of the random walks has "more randomness" than the other. We also treat some related models: The parabolic Anderson model with space-time Lévy noise; Brownian motion among space-time obstacles; and branching random walks in space-time random environments. We also obtain a necessary and sufficient criterion for $Z^1\preceq_{cv}Z^2$ if the lattice is replaced by a regular tree.

math.PR↗
arXiv2017

A branching random walk among disasters

We consider a branching random walk in a random space-time environment of disasters where each particle is killed when meeting a disaster. This extends the model of the "random walk in a disastrous random environment" introduced by [15]. We obtain a criterion for positive survival probability, see Theorem 1. The proofs for the subcritical and the supercritical cases follow standard arguments, which involve moment methods and a comparison with an embedded branching process with i.i.d. offspring distributions. The proof of almost sure extinction in the critical case is more difficult and uses the techniques from [8]. We also show that, in the case of survival, the number of particles grows exponentially fast.

math.PR↗
arXiv2015

On the survival probability of a random walk in random environment with killing

We consider one dimensional random walks in random environment where every time the process stays at a location, it dies with a fixed probability. Under some mild assumptions it is easy to show that the survival probability goes to zero as time tends to infinity. In this paper we derive formulas for the rate with which this probability decays. It turns out that there are three distinct regimes, depending on the law of the environment.

math.PR↗