# Celebrating the new probability group

## Random permutations and large deviations for Markov chains

### Wednesday 15th and Thursday 16th June 2016

The object of this meeting is to celebrate the recent expansion in the probability group in the department. The organisers, Dmitry Korshunov and Dirk Zeindler, would like to use this opportunity to share their research interests with their colleagues, including the statistics section. Also, it is our aim to use this event to raise current issues about random permutations, to discuss various aspects of limit theorems and large deviations for Markov chains and to discuss new approaches and new questions. This meeting is funded by the London Mathematical Society (LMS) and everybody is welcome to take part.

## Schedule

Wednesday 15th

- 11:00 - 12:00: Dirk Zeindler
- 14:15 - 15:15: Daniel Ueltschi
- 15:30 - 16:30: Leonid Bogachev
- 18:00: Dinner

Thursday 16th

- 11:00 - 12:00: Denis Denisov
- 14:00 - 15:00: Dmitry Korshunov
- 15:30 - 16:30: Sergey Foss

The local organising committee consists of Dmitry Korshunov and Dirk Zeindler.

Please direct any enquiries to the organisers or to probability@lancaster.ac.uk.

## Speakers and Abstracts

### Leonid Bogachev

#### Large Components in Random Decomposable Structures, and Bose–Einstein Condensation

Random decomposable structures of big size (such as integer partitions or permutations) may manifest emergence of “large” components (or even a single giant component) due to asymptotic loss of mass. A. M. Vershik, inﬂuenced by discussions with R. L. Dobrushin, R. A. Minlos and Ya. G. Sinai, was probably the ﬁrst to observe (around 1995–1997) a remarkable link between random integer partitions and models of statistical mechanics; in particular, he has pointed out an interpretation of the limit shape (of associated Young diagrams) in terms of the celebrated Bose–Einstein condensation of ideal quantum gas. Recently, there have been many eﬀorts to elaborate these connections in order to make progress in understanding and explaining Bose–Einstein condensation for the interacting gas. In this talk, I will present two recent results in this direction, one related to a uniﬁed derivation of the limit shape of random partitions (DOI:10.1002/rsa.20540; arXiv:1111.3311), and the other one (joint work with Dirk Zeindler) about the asymptotic statistics of cycles in the model of “surrogate-spatial” permutations (DOI:10.1007/s00220-014-2110-1; arXiv:1309.7986), proposed as an analytically tractable version of the model of spatial random permutations extensively studied in the physical context in a series of papers by V. Betz and D. Ueltschi.

### Denis Denisov

#### Heavy-traffic and heavy tails for random walks

Consider a family of random walks Sn(a) with negative drift E[Sn(a)] =-a<0 and finite variance. Let Mn(a) =maxn≥0 Sn(a) be the maximum of the random walk. It is known that the probability P(Mn(a)>x) decays exponentially fast as a →0 (heavy traffic asymptotics) and, for subexponential distributions P(Mn(a)>x) decays according to the integrated tail as x→∞. We will present a link between this two asymptotics and study the probability P(Mn(a)>x) and identify the regions of x for which the heavy traffic asymptotics and the heavy tail asymptotics hold.

### Sergey Foss

#### On a limiting behaviour of a conditional random walk with bounded local times

Joint work with Alexander Sakhanenko (Novosibirsk) We consider a random walk {X_n} on the integers Z= {0, ±1, ±2,...} with transition probabilities pk = P(Xn+1 - Xn=k) where p1>0, p1+p0<1 and pk=0, for k>1. Suppose X0=0. Further, assume each state j \in Z is initially given L >2 units of energy and each time when the random walk visits a state, it takes 1 unit of energy to leave it. Eventually, such a random walk arrives (with probability 1) to a state with no energy -- then it "freezes" there. For any state j, let T(j) be the first time the random walk visits j and let T(j) be infinite it j is never visited. Clearly, each T(j) takes the infinite value with a positive probability, say r_j. Moreover, rj tends to 1 as the absolute value |j| tends to infinity. We consider the trajectory of {X_n} on the time interval 0,1,..., T(j) conditionally on the event that T(j) is finite. We show that, as j tends to infinity, this trajectory converges in a "strong" sense to a trajectory of a regenerative process. We also discuss a number of related problems. This work is motivated by a paper by I. Benjamini and N. Berestycki (2010) where the case of a simple symmetric random walk has been considered.

### Dmitry Korshunov

#### Stability and instability of diffusion processes and Markov chains with asymptotically zero drift

Study of transience, positive and null recurrence of Markov chains with asymptotically zero is often referred to as Lamperti's problem. Motivated by necessary and sufficient conditions for transience, positive and null recurrence available for diffusions, we suggest their analogues for Markov chains. In this way, we significantly improve various conditions known in the literature for Markov chains. Joint work with Denis Denisov (Manchester) and Vitali Wachtel (Augsburg).

### Daniel Ueltschi

#### Random loop models and quantum spin systems

The random loop representations of quantum Heisenberg models allow to study these systems using probabilistic methods. They were introduced twenty years ago by Toth and Aizenman-Nachtergaele. I will review rigorous results and open questions. I will describe a probabilistic proof of exponential decay of transverse quantum correlations (joint work with J. Bjornberg).

### Dirk Zeindler

#### The order of large random permutations with cycle weights

The order On of a permutation of n objects is the smallest integer k such that the k-th iterate of gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdos-Turan who proved in 1965 that log On satisfies a central limit theorem. We show that the Erdos-Turan Law can be extended to random permutations chosen according to the so-called generalized Ewens measure and to a generalized weighted measure with polynomially growing cycle weights. Furthermore, we establish for the generalized Ewens measure a local limit theorem as well as, under some extra moment condition, a precise large deviation estimate and also show that the expectation of the logarithm of the order has a remarkable connection with the Riemann hypothesis. In addition, we provide a precise large deviation estimate for random permutations with polynomial growing cycle weights.