Attempted Bethe ansatz solution for one-dimensional directed polymers in random media

We study the statistical properties of one-dimensional directed polymers in a short-range random potential by mapping the replicated problem to a many body quantum boson system with attractive interactions. We find the full set of eigenvalues and eigenfunctions of the many-body system and perform the summation over the entire spectrum of excited states. The analytic continuation of the obtained exact expression for the replica partition function from integer to non-integer replica parameter N turns out to be ambiguous. Performing the analytic continuation simply by assuming that the parameter N can take arbitrary complex values, and going to the thermodynamic limit of the original directed polymer problem, we obtain the explicit universal expression for the probability distribution function of free energy fluctuations.Comment: 32 pages, 1 figur

One loop partition function in AdS_3/CFT_2

The 1-loop partition function of the handle-body solutions in the AdS$_3$ gravity have been derived some years ago using the heat-kernel and the method of images. In the semiclassical limit, such partition function should correspond to the order $O (c^0)$ part in the partition function of dual conformal field theory on the boundary Riemann surface. The higher genus partition function could be computed by the multi-point functions in the Riemann sphere via sewing prescription. In the large central charge limit, to the leading order of $c$, the multi-point function is further simplified to be a summation over the product of two-point functions, which may form links. Each link is in one-to-one correspondence with the conjugacy class of the Schottky group of the Riemann surface. Moreover, the value of a link is determined by the eigenvalue of the element in the conjugate class. This allows us to reproduce exactly the gravitational 1-loop partition function. The proof can be generalized to the higher spin gravity and its dual CFT.Comment: 30 pages, 8 figures; typos corrected, more clarifications, references and acknowledgements adde

Renormalization group approach to chaotic strings

Coupled map lattices of weakly coupled Chebychev maps, so-called chaotic strings, may have a profound physical meaning in terms of dynamical models of vacuum fluctuations in stochastically quantized field theories. Here we present analytic results for the invariant density of chaotic strings, as well as for the coupling parameter dependence of given observables of the chaotic string such as the vacuum expectation value. A highly nontrivial and selfsimilar parameter dependence is found, produced by perturbative and nonperturbative effects, for which we develop a mathematical description in terms of suitable scaling functions. Our analytic results are in good agreement with numerical simulations of the chaotic dynamics.Comment: 36 pages, 18 figures - v2 contains slightly more than the published versio

Computing Exact Clustering Posteriors with Subset Convolution

An exponential-time exact algorithm is provided for the task of clustering n items of data into k clusters. Instead of seeking one partition, posterior probabilities are computed for summary statistics: the number of clusters, and pairwise co-occurrence. The method is based on subset convolution, and yields the posterior distribution for the number of clusters in O(n * 3^n) operations, or O(n^3 * 2^n) using fast subset convolution. Pairwise co-occurrence probabilities are then obtained in O(n^3 * 2^n) operations. This is considerably faster than exhaustive enumeration of all partitions.Comment: 6 figure

Spin glass reflection of the decoding transition for quantum error correcting codes

We study the decoding transition for quantum error correcting codes with the help of a mapping to random-bond Wegner spin models. Families of quantum low density parity-check (LDPC) codes with a finite decoding threshold lead to both known models (e.g., random bond Ising and random plaquette $\Z2$ gauge models) as well as unexplored earlier generally non-local disordered spin models with non-trivial phase diagrams. The decoding transition corresponds to a transition from the ordered phase by proliferation of extended defects which generalize the notion of domain walls to non-local spin models. In recently discovered quantum LDPC code families with finite rates the number of distinct classes of such extended defects is exponentially large, corresponding to extensive ground state entropy of these codes. Here, the transition can be driven by the entropy of the extended defects, a mechanism distinct from that in the local spin models where the number of defect types (domain walls) is always finite.Comment: 15 pages, 2 figure

On the duality relation for correlation functions of the Potts model

We prove a recent conjecture on the duality relation for correlation functions of the Potts model for boundary spins of a planar lattice. Specifically, we deduce the explicit expression for the duality of the n-site correlation functions, and establish sum rule identities in the form of the M\"obius inversion of a partially ordered set. The strategy of the proof is by first formulating the problem for the more general chiral Potts model. The extension of our consideration to the many-component Potts models is also given.Comment: 17 pages in RevTex, 5 figures, submitted to J. Phys.

Making simple proofs simpler

An open partition \pi{} [Cod09a, Cod09b] of a tree T is a partition of the vertices of T with the property that, for each block B of \pi, the upset of B is a union of blocks of \pi. This paper deals with the number, NP(n), of open partitions of the tree, V_n, made of two chains with n points each, that share the root

Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers

The distribution function of the free energy fluctuations in one-dimensional directed polymers with $\delta$-correlated random potential is studied by mapping the replicated problem to the $N$-particle quantum boson system with attractive interactions. We find the full set of eigenfunctions and eigenvalues of this many-body system and perform the summation over the entire spectrum of excited states. It is shown that in the thermodynamic limit the problem is reduced to the Fredholm determinant with the Airy kernel yielding the universal Tracy-Widom distribution, which is known to describe the statistical properties of the Gaussian unitary ensemble as well as many other statistical systems.Comment: 23 page