Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ on some fixed time interval $[0,β]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $σ(i)$-th motion, where $σ$ is a uniformly distributed random permutation of $1,...,N$.
In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the $N$ paths) when $ Λ\uparrow\mathbb{R}^d $ and $ N/|Λ|\toρ$. The rate function is given as a variational formula involving a certain entropy functional and a Fenchel-Legendre transform.
Depending on the dimension and the density $ ρ$, there is phase transition behaviour for the empirical path measure. For certain parameters (high density, large time horizon) and dimensions $ d\ge 3 $ the empirical path measure is not supported on all paths $ [0,\infty)\to\mathbb{R}^d $ which contain a bridge path of any finite multiple of the time horizon $ [0,β] $. For dimensions $ d=1,2 $, and for small densities and small time horizon $ [0,β] $ in dimensions $ d\ge 3$, the empirical path measure is supported on those paths. In the first regime a finite fraction of the motions lives in cycles of infinite length.
We outline that this transition leads to an empirical path measure interpretation of {\it Bose-Einstein condensation}, known for systems of Bosons.
Here we introduce a family of spatial cycle HYL-type models in which the counter-term only affects cycles longer than some cut-off that diverges in the thermodynamic limit. We derive large deviation principles and explicit pressure expressions for these models, and use the zeroes of the rate functions to study Bose-Einstein condensation.
We prove moderate deviations bounds for the lower tail of the number of odd cycles in a $\calG(n, m)$ random graph. We show that the probability of decreasing triangle density by $t^3$, is $\exp(-Θ(n^2 t^2))$ whenever $n^{-3/4} \ll t^3 \ll 1$, while for $k \ge 5$ we give the same estimate for the probability of decreasing the $k$-cycle density by $t^k$, but for the larger range $n^{-1} \ll t^k \ll 1$. When $m \ge \frac 12 \binom n2$, we also find the leading coefficient in the exponent. This complements results of Goldschmidt et al., who showed that for $n^{-3/2} \ll t^k \ll n^{-1}$, the proba…
We consider a continuous time Markov chain on a countable state space. We prove a joint large deviation principle (LDP) of the empirical measure and current in the limit of large time interval. The proof is based on results on the joint large deviations of the empirical measure and flow obtained in \cite{BFG}. By improving such results we also show, under additional assumptions, that the LDP holds with the strong L^1 topology on the space of currents. We deduce a general version of the Gallavotti-Cohen (GC) symmetry for the current field and show that it implies the so-called fluctuation theor…
We prove a large deviations principle for the empirical law of the block sizes of a uniformly distributed non-crossing partition. As an application we obtain a variational formula for the maximum of the support of a compactly supported probability measure in terms of its free cumulants, provided these are all non-negative. This is useful in free probability theory, where sometimes the R-transform is known but cannot be inverted explicitly to yield the density.
Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the two-parameter Poisson-Dirichlet distribution. A sample of size n is selected from the population. The total number of different types and the number of types appearing in the sample with a fixed frequency are important statistics. In this paper we establish the moderate deviation principles for these quantities. The corresponding rate functions are explicitly identified, which help revealing a critical scale and understanding the exact role of the parameters. Conditional, …
We study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials. The main purpose is the derivation of explicit formulas for the $1+1$-dimensional exactly solvable case with log-gamma distributed random weights. Along the way we establish some regularity results for this rate function for general distributions in arbitrary dimensions.
Using Talagrand's concentration inequality on the discrete cube {0,1}^m we show that given a real-valued function Z(x)on {0,1}^m that satisfies certain monotonicity conditions one can control the deviations of Z(x) above its median by a local Lipschitz norm of Z(x) at the point x. As one application, we give a simple proof of a nearly optimal deviation inequality for the number of k-cycles in a random graph.
This paper studies probabilistic mean-field models for interacting bosons at a positive temperature in the thermodynamic limit with random particle density. In particular, we prove large deviation principles for empirical cycle counts in all our models, and as such generalise recent work in \cite{ACK} where upper and lower large deviation bounds do not match. Namely, on the one hand we generalise to the so-called grand canonical ensemble, and on the other hand, we consider classes of interaction potentials depending on the cycle counts which are not restricted to be positive. Our large deviati…
We prove general nonlinear large deviation estimates similar to Chatterjee-Dembo's original bounds except that we do not require any second order smoothness. Our approach relies on convex analysis arguments and is valid for a broad class of distributions. Our results are then applied in three different setups. Our first application consists in the mean-field approximation of the partition function of the Ising model under an optimal assumption on the spectra of the adjacency matrices of the sequence of graphs. Next, we apply our general large deviation bound to investigate the large deviation …
Consider the partition function of a directed polymer in an IID field. We assume that both tails of the negative and the positive part of the field are at least as light as exponential. It is a well-known fact that the free energy of the polymer is equal to a deterministic constant for almost every realization of the field and that the upper tail of the large deviations is exponential. The lower tail of the large deviations is typically lighter than exponential. In this paper we provide a method to obtain estimates on the rate of decay of the lower tail of the large deviations, which are sharp…
In this paper, we study various models for random combinatorial partitions using large deviation analysis for diverging scale of the reference process. Scaling limits of similar models have been studied recently \cite{FSa,FSb} going back to \cite{Ver96}. After studying the reference model, we provide a complete analysis of two mean field models, one of which is well-know \cite{BCMP05} and the other one is the cycle mean field model. Both models show critical behaviour despite their rate functions having unique minimiser. The main focus is then a model with negative counter term, the probabilis…
Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds: cycle covers, closed walks, spanning trees. We show that if the weights of the edges of the graph are within a constant factor, fixed in advance, of each other then the bulk of the partition function is concentrated on the subsets of a particularly simple structure: cycle covers with few cycles, walks that visit every vertex only few times, and spanning trees wi…
In this short note we consider semi-Markov processes satisfying the condition of direction-time independence (Markov renewal processes). We derive large deviation principles and fluctuation theorems for the empirical current and the empirical currents along cycles. Our derivation is based on the joint LDP for the empirical measure and flow recently proved in \cite{MZ}.
We consider a system of classical particles confined in a box $Λ\subset\mathbb{R}^d$ with zero boundary conditions interacting via a stable and regular pair potential. Based on the validity of the cluster expansion for the canonical partition function in the high temperature - low density regime we prove moderate and precise large deviations from the mean value of the number of particles with respect to the grand-canonical Gibbs measure. In this way we have a direct method of computing both the exponential rate as well as the pre-factor and obtain explicit error terms. Estimates comparing with…
We study random surfaces constructed by glueing together $N/k$ filled $k$-gons along their edges, with all $(N-1)!! = (N-1)(N-3)...3\cdot 1$ pairings of the edges being equally likely. (We assume that lcm $\{2,k\}$ divides $N$.) The Euler characteristic of the resulting surface is related to the number of cycles in a certain random permutation of $\{1, ..., N\}$. Gamburd has shown that when 2 lcm $\{2,k\}$ divides $N$, the distribution of this random permutation converges to that of the uniform distribution on the alternating group $A_N$ in the total-variation distance as $N\to\infty$. We obta…
We consider random partitions of the vertex set of a given finite graph that can be sampled by means of loop-erased random walks stopped at a random exponential time of parameter $q>0$. The related random blocks tend to cluster nodes visited by the random walk on time scale $1/q$. This random partitioning is induced by a measure of rooted spanning forest of the graph, which generalizes the classical uniform spanning tree measure and which can be obtained as a zero-limit of FK-percolation with an external cemetery state. Some general properties of this rooted forest measure and related determin…
The order $O_n(σ)$ of a permutation $σ$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $σ$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdös and Turán who proved in 1965 that $\log O_n$ satisfies a central limit theorem. We extend this result to the so-called \textit{generalized Ewens measure} in a previous paper. In this paper, we establish a local limit theorem as well as, under some extra moment condition, a precise large deviations estimate. These properties are new even for the uniform measure. Furt…
Problem 1.5.7 from Pitman's Saint-Flour lecture notes: Does there exist for each n a fragmentation process (Π_{n,k}, 1 \leq k \leq n) taking values in the space of partitions of {1,2,...,n} such that Π_{n,k} is distributed like the partition generated by cycles of a uniform random permutation of {1,2,...,n} conditioned to have k cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions.
In probability theory, equalities are much less than inequalities. In this paper, we find a series of equalities which characterize the symmetry of the forming times of a family of similar cycles for discrete-time and continuous-time Markov chains. Moreover, we use these cycle symmetries to study the circulation fluctuations for Markov chains. We prove that the empirical circulations of a family of cycles passing through a common state satisfy a large deviation principle with a rate function which has an highly non-obvious symmetry. Finally, we discuss the applications of our work in statistic…
