Cutoff profile of ASEP on a segment

This paper studies the mixing behavior of the Asymmetric Simple Exclusion Process (ASEP) on a segment of length N. Our main result is that for particle densities in (0, 1), the total-variation cutoff window of ASEP is $$N^{1/3}$$ N 1 / 3 and the cutoff profile is $$1-F_{\mathrm {GUE}},$$ 1 - F GUE , where $$F_{\mathrm {GUE}}$$ F GUE is the Tracy-Widom distribution function. This also gives a new proof of the cutoff itself, shown earlier by Labbé and Lacoin. Our proof combines coupling arguments, the result of Tracy–Widom about fluctuations of ASEP started from the step initial condition, and exact algebraic identities coming from interpreting the multi-species ASEP as a random walk on a Hecke algebra.

Limiting Directions for Random Walks in Classical Affine Weyl Groups

Let $W$ be a finite Weyl group and $\widetilde W$ the corresponding affine Weyl group. A random element of $\widetilde W$ can be obtained as a reduced random walk on the alcoves of $\widetilde W$. By a theorem of Lam (Ann. Prob. 2015), such a walk almost surely approaches one of $|W|$ many directions. We compute these directions when $W$ is $B_n$, $C_n$, and $D_n$ and the random walk is weighted by Kac and dual Kac labels. This settles Lam’s questions for types $B$ and $C$ in the affirmative and for type $D$ in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process (TASEP) called the $D^*$-TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.

Integrability of the Multi-Species TASEP with Species-Dependent Rates

Assume that each species l has its own jump rate bl in the multi-species totally asymmetric simple exclusion process. We show that this model is integrable in the sense that the Bethe ansatz method is applicable to obtain the transition probabilities for all possible N-particle systems with up to N different species.

Mapping TASEP back in time

We obtain a new relation between the distributions $$\upmu _t$$ μ t at different times $$t\ge 0$$ t ≥ 0 of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions $$\upmu _t$$ μ t backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving $$\upmu _t$$ μ t which in turn brings new identities for expectations with respect to $$\upmu _t$$ μ t . The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions.

The Yellow Active Queue Management Algorithm in ICN Routers Based on the Monitoring of Bandwidth Competition

Deploying the active queue management (AQM) algorithm on a router is an effective way to avoid packet loss caused by congestion. In an information-centric network (ICN), routers not only play a role of packets forwarding but are also content service providers. Congestion in ICN routers can be further summarized as the competition between the external forwarding traffic and the internal cache response traffic for limited bandwidth resources. This indicates that the traditional AQM needs to be redesigned to adapt to ICN. In this paper, we first demonstrated mathematically that allocating more bandwidth for the upstream forwarding flow could improve the quality of service (QoS) of the whole network. Secondly, we propose a novel AQM algorithm, YELLOW, which predicts the bandwidth competition event and adjusts the input rate of request and the marking probability adaptively. Afterwards, we model YELLOW through the totally asymmetric simple exclusion process (TASEP) and deduce the approximate solution of the existence condition for each stationary phase. Finally, we evaluated the performance of YELLOW by NS-3 simulator, and verified the accuracy of modeling results by Monte Carlo. The simulation results showed that the queue of YELLOW could converge to the expected value, and the significant gains of the router with low packet loss rate, robustness and high throughput.

Mapping current and activity fluctuations in exclusion processes: consequences and open questions

Considering the large deviations of activity and current in the Asymmetric Simple Exclusion Process (ASEP), we show that there exists a non-trivial correspondence between the joint scaled cumulant generating functions of activity and current of two ASEPs with different parameters. This mapping is obtained by applying a similarity transform on the deformed Markov matrix of the source model in order to obtain the deformed Markov matrix of the target model. We first derive this correspondence for periodic boundary conditions, and show in the diffusive scaling limit (corresponding to the Weakly Asymmetric Simple Exclusion Processes, or WASEP) how the mapping is expressed in the language of Macroscopic Fluctuation Theory (MFT). As an interesting specific case, we map the large deviations of current in the ASEP to the large deviations of activity in the SSEP, thereby uncovering a regime of Kardar--Parisi--Zhang in the distribution of activity in the SSEP. At large activity, particle configurations exhibit hyperuniformity [Jack et al., PRL 114, 060601 (2015)]. Using results from quantum spin chain theory, we characterize the hyperuniform regime by evaluating the small wavenumber asymptotic behavior of the structure factor at half-filling. Conversely, we formulate from the MFT results a conjecture for a correlation function in spin chains at any fixed total magnetization (in the thermodynamic limit). In addition, we generalize the mapping to the case of two open ASEPs with boundary reservoirs, and we apply it in the WASEP limit in the MFT formalism. This mapping also allows us to find a symmetry-breaking dynamical phase transition (DPT) in the WASEP conditioned by activity, from the prior knowledge of a DPT in the WASEP conditioned by the current.