A product formula for the TASEP on a ring

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

2018

Self Cite

In this paper, we study a distribution Ξ of labeled particles on a continuous ring. It arises in three different ways, all related to the multi-type TASEP on a ring. We prove formulas for the probability density function for some permutations and give conjectures for a larger class. We give a complete conjecture for the probability of two particles i, j being next to each other on the cycle, for which we prove some cases. We also find that two natural events associated to the process have exactly the same probability expressed as a Vandermonde determinant. It is unclear whether this is just a coincidence or a consequence of a deeper connection.2000 Mathematics Subject Classification. 60K35,82C22.

Section: Introductionmentioning

confidence: 74%

Continuous multi-line queues and TASEP

Aas

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

2018

Self Cite

“…Therefore, there is one x 2 transition for every block of 3's, the leftmost of which is not covered and no other x 2 transitions. Now, let us look at incoming transitions in the master equation (1). It will be convenient to use the notion of effective rate defined in equation (2).…”

Section: Three Species Ferrari-martin Processmentioning

confidence: 99%

“…There is thus a contribution of (x 1 + k − 1)w(Q) to the outgoing transitions from Q. Now, let us look at incoming transitions to Q in the master equation (1). The key observation is illustrated by Figure 5, which is a cartoon for a configuration Q along with a generic ringing path transition at j to Q. j Figure 5.…”

Section: Zmmentioning

confidence: 99%

“…Note added in proof: Since the submission of the paper there has been progress and there are now claimed proofs of both the formula for the stationary weight of the identity by Aas [1] and for our Conjecture 2.10 (two very different ways by Arita-Malik [3] and Linusson-Martin [18]). …”

Section: Introductionmentioning

confidence: 99%

“…Now to the incoming side of the master equation (1). We will again use the notion of effective rate (see (2)).…”

mentioning

confidence: 99%

See 2 more Smart Citations

An inhomogeneous multispecies TASEP on a ring

Ayyer

Advances in Applied Mathematics

2014

Abstract. We reinterpret and generalize conjectures of Lam and Williams as statements about the stationary distribution of a multispecies exclusion process on the ring. The central objects in our study are the multiline queues of Ferrari and Martin. We make some progress on some of the conjectures in different directions. First, we prove Lam and Williams' conjectures in two special cases by generalizing the rates of the Ferrari-Martin transitions. Secondly, we define a new process on multiline queues, which have a certain minimality property. This gives another proof for one of the special cases; namely arbitrary jump rates for three species.

Correlations in the multispecies TASEP and a conjecture by Lam

Ayyer

Trans. Amer. Math. Soc.

2016

We study correlations in the multispecies TASEP on a ring. Results on correlation of two adjacent points prove two conjectures by Thomas Lam on (a) the limiting direction of a reduced random walk inÃn−1 and (b) the asymptotic shape of a random integer partition with no hooks of length n, a so called n-core.We further investigate two-point correlations far apart and threepoint nearest neighbour correlations and prove explicit formulas in almost all cases. These results can be seen as a finite strengthening of correlations in the TASEP speed process by Amir, Angel and Valkó. We also give conjectures for certain higher order nearest neighbour correlations. We find an unexplained independence property (provably for two points, conjecturally for more points) between points that are closer in position than in value that deserves more study.