Random walk on surfaces with hyperbolic cusps

We consider the operator associated to a random walk on finite volume surfaces with hyperbolic cusps. We study the spectral gap (upper and lower bound) associated to this operator and deduce some rate of convergence of the iterated kernel towards its stationary distribution.Comment: 28 page

Cutoff for the Ising model on the lattice

Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in $L^1$ on a system of size $n$ is $O(\log n)$. Whether in this regime there is cutoff, i.e. a sharp transition in the $L^1$-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at $(c+o(1))\log n$ for some fixed $c>0$, thus providing a rigorous stopping rule for this MCMC sampler. However, obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple Markov chains. Already for the one-dimensional Ising model, showing cutoff is a longstanding open problem. We settle the above by establishing cutoff and its location at the high temperature regime of the Ising model on the lattice with periodic boundary conditions. Our results hold for any dimension and at any temperature where there is strong spatial mixing: For $\Z^2$ this carries all the way to the critical temperature. Specifically, for fixed $d\geq 1$, the continuous-time Glauber dynamics for the Ising model on $(\Z/n\Z)^d$ with periodic boundary conditions has cutoff at $(d/2\lambda_\infty)\log n$, where $\lambda_\infty$ is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited. The proof hinges on a new technique for translating $L^1$ to $L^2$ mixing which enables the application of log-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems.Comment: 34 pages, 3 figure

Entropy-driven cutoff phenomena

In this paper we present, in the context of Diaconis' paradigm, a general method to detect the cutoff phenomenon. We use this method to prove cutoff in a variety of models, some already known and others not yet appeared in literature, including a chain which is non-reversible w.r.t. its stationary measure. All the given examples clearly indicate that a drift towards the opportune quantiles of the stationary measure could be held responsible for this phenomenon. In the case of birth- and-death chains this mechanism is fairly well understood; our work is an effort to generalize this picture to more general systems, such as systems having stationary measure spread over the whole state space or systems in which the study of the cutoff may not be reduced to a one-dimensional problem. In those situations the drift may be looked for by means of a suitable partitioning of the state space into classes; using a statistical mechanics language it is then possible to set up a kind of energy-entropy competition between the weight and the size of the classes. Under the lens of this partitioning one can focus the mentioned drift and prove cutoff with relative ease.Comment: 40 pages, 1 figur

Cutoff for the East process

The East process is a 1D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on $L$ sites has order $L$. We complement that result and show cutoff with an $O(\sqrt{L})$-window. The main ingredient is an analysis of the front of the process (its rightmost zero in the setup where zeros facilitate updates to their right). One expects the front to advance as a biased random walk, whose normal fluctuations would imply cutoff with an $O(\sqrt{L})$-window. The law of the process behind the front plays a crucial role: Blondel showed that it converges to an invariant measure $\nu$, on which very little is known. Here we obtain quantitative bounds on the speed of convergence to $\nu$, finding that it is exponentially fast. We then derive that the increments of the front behave as a stationary mixing sequence of random variables, and a Stein-method based argument of Bolthausen ('82) implies a CLT for the location of the front, yielding the cutoff result. Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an $O(1)$-window.Comment: 33 pages, 2 figure

Discrete analogue computing with rotor-routers

Rotor-routing is a procedure for routing tokens through a network that can implement certain kinds of computation. These computations are inherently asynchronous (the order in which tokens are routed makes no difference) and distributed (information is spread throughout the system). It is also possible to efficiently check that a computation has been carried out correctly in less time than the computation itself required, provided one has a certificate that can itself be computed by the rotor-router network. Rotor-router networks can be viewed as both discrete analogues of continuous linear systems and deterministic analogues of stochastic processes.Comment: To appear in Chaos Special Focus Issue on Intrinsic and Designed Computatio

Stein's Method and Characters of Compact Lie Groups

Stein's method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the square of the trace into irreducible components. This is illustrated for Lie groups of classical type and Dyson's circular ensembles. The approach in this paper will be useful for the study of higher dimensional characters, where normal approximations need not hold.Comment: 22 pages; same results, but more efficient exposition in Section 3.

Superclasses and supercharacters of normal pattern subgroups of the unipotent upper triangular matrix group

Let $U_n$ denote the group of $n\times n$ unipotent upper-triangular matrices over a fixed finite field \FF_q, and let U_\cP denote the pattern subgroup of $U_n$ corresponding to the poset \cP. This work examines the superclasses and supercharacters, as defined by Diaconis and Isaacs, of the family of normal pattern subgroups of $U_n$. After classifying all such subgroups, we describe an indexing set for their superclasses and supercharacters given by set partitions with some auxiliary data. We go on to establish a canonical bijection between the supercharacters of U_\cP and certain \FF_q-labeled subposets of \cP. This bijection generalizes the correspondence identified by Andr\'e and Yan between the supercharacters of $U_n$ and the \FF_q-labeled set partitions of $\{1,2,...,n\}$. At present, few explicit descriptions appear in the literature of the superclasses and supercharacters of infinite families of algebra groups other than \{U_n : n \in \NN\}. This work signficantly expands the known set of examples in this regard.Comment: 28 page

A quantum de Finetti theorem in phase space representation

The quantum versions of de Finetti's theorem derived so far express the convergence of n-partite symmetric states, i.e., states that are invariant under permutations of their n parties, towards probabilistic mixtures of independent and identically distributed (i.i.d.) states. Unfortunately, these theorems only hold in finite-dimensional Hilbert spaces, and their direct generalization to infinite-dimensional Hilbert spaces is known to fail. Here, we address this problem by considering invariance under orthogonal transformations in phase space instead of permutations in state space, which leads to a new type of quantum de Finetti's theorem that is particularly relevant to continuous-variable systems. Specifically, an n-mode bosonic state that is invariant with respect to this continuous symmetry in phase space is proven to converge towards a probabilistic mixture of i.i.d. Gaussian states (actually, n identical thermal states).Comment: 5 page

Relaxation due to random collisions with a many-qudit environment

We analyze the dynamics of a system qudit of dimension mu sequentially interacting with the nu-dimensional qudits of a chain playing the ore of an environment. Each pairwise collision has been modeled as a random unitary transformation. The relaxation to equilibrium of the purity of the system qudit, averaged over random collisions, is analytically computed by means of a Markov chain approach. In particular, we show that the steady state is the one corresponding to the steady state for random collisions with a single environment qudit of effective dimension nu_e=nu*mu. Finally, we numerically investigate aspects of the entanglement dynamics for qubits (mu=nu=2) and show that random unitary collisions can create multipartite entanglement between the system qudit and the qudits of the chain.Comment: 7 pages, 6 figure