Nonequilibrium Luttinger liquid: zero-bias anomaly and dephasing

A one-dimensional system of interacting electrons out of equilibrium is studied in the framework of the Luttinger liquid model. We analyze several setups and develop a theory of tunneling into such systems. A remarkable property of the problem is the absence of relaxation in energy distribution functions of left and right movers yet the presence of the finite dephasing rate due to electron-electron scattering, which smears zero-bias-anomaly singularities in the tunneling density of states.     

World spinors—Construction and some applications

The existence of a topological double-covering for the GL(n, R) and diffeomorphism groups is reviewed. These groups do not have finite-dimensional faithful representations. An explicit construction and the classification of all (n, R), n=3,4 unitary irreducible representations is presented. Infinite-component spinorial and tensorial fields, “manifields”, are introduced. Particle content of the ladder manifields, as given by the (3, R) “little” group, is determined. The manifields are lifted to the corresponding world spinorial and tensorial manifields by making use of generalized infinite-component frame fields. World manifields transform w.r.t. corresponding representations, which are constructed explicitly. Supported in part by the Science Foundation (Belgrade).     

The Sign and Micro-Origin of Complexity as Entropy (Conjectured Resolution of a Paradox)

Evolution produces ever more ordered matter, while also increasing its complexity all the time. There are various ways of measuring complexity, such as Kolmogorov's algorithmic complexity, drawn from information theory, and identified with entropy, enchancing irreversibility in harmony with the second law of thermodynamics. On the other hand, however, the creation of order should have reduced entropy; quoting Schroedinger, it represents "negentropy." To resolve this apparent contradiction we first review a similar set up (though with a totally different interaction) occurring in black holes, a model in which the physics are now explicit and fully understood at the quantum level.     

Two-soliton interaction in the vicinity of a defect inside a fiber Bragg grating and its application for obtaining an all-optical memory

We study the interaction between two Bragg solitons in the vicinity of a defect inside a fiber Bragg grating. A soliton that is trapped in the defect can be released by launching a second soliton. The effect can be used to obtain an all-optical memory that is not strongly sensitive to the phase and the timing arrival of the solitons.     

Evolving sets, mixing and heat kernel bounds

We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovász and Kannan, can be refined to apply to the maximum relative deviation |pⁿ(x,y)/π(y)−1| of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.Research supported in part by NSF Grants #DMS-0104073 and #DMS-0244479.     

Bounds for resultants of univariate and bivariate polynomials

The paper considers bounds on the size of the resultant for univariate and bivariate polynomials. For univariate polynomials we also extend the traditional representation of the resultant by the zeros of the argument polynomials to formal resultants, defined as the determinants of the Sylvester matrix for a pair of polynomials whose actual degree may be lower than their formal degree due to vanishing leading coefficients. For bivariate polynomials, the resultant is a univariate polynomial resulting by the elimination of one of the variables, and our main result is a bound on the largest coefficient of this univariate polynomial. We bring a simple example that shows that our bound is attainable and that a previous sharper bound is not correct.     

Critical percolation on random regular graphs

The behavior of the random graph G(n,p) around the critical probability p_c = is well understood. When p = (1 + O(n^1/3))p_c the components are roughly of size n^2/3 and converge, when scaled by n^?2/3, to excursion lengths of a Brownian motion with parabolic drift. In particular, in this regime, they are not concentrated. When p = (1 ‐ ?(n))p_c with ?(n)n^1/3 →∞ (the subcritical regime) the largest component is concentrated around 2?^?2 log(?³n). When p = (1 + ?(n))p_c with ?(n)n^1/3 →∞ (the supercritical regime), the largest component is concentrated around 2?n and a duality principle holds: other component sizes are distributed as in the subcritical regime. Itai Benjamini asked whether the same phenomenon occurs in a random d‐regular graph. Some results in this direction were obtained by (Pittel, Ann probab 36 (2008) 1359–1389). In this work, we give a complete affirmative answer, showing that the same limiting behavior (with suitable d dependent factors in the non‐critical regimes) extends to random d‐regular graphs. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie–Weiss Model. For β < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1 ? β)]^?1 n log n. For β = 1, we prove that the mixing time is of order n ^3/2. For β > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).     

On the sum of the L1 influences of bounded functions

Let f: {-1, 1}ⁿ → [-1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L₁ influence of f can also be bounded as a function of d. Ba?kurs and Bavarian answered this question in the affirmative, providing a bound of O(d³) for general functions and O(d²) for homogeneous functions. We improve on their results by providing a bound of d² for general functions and O(d log d) for homogeneous functions. In addition, we prove a bound of d/(2p) + o(d) for monotone functions, and provide a matching example.     

Upper bound on the characters of the symmetric groups

Let C be a conjugacy class in the symmetric group S _n , and λ be a partition of n. Let f ^λ be the degree of the irreducible representation S ^λ , χ ^λ (C)– the character of S ^λ at C, and r ^λ (C)– the normalized character χ ^λ (C) f ^λ . We prove that there exist constants b > 0 and 1 > q > 0 such that for n > 4, for every conjugacy class C in S _n and every irreducible representation S ^λ of S _n ∣r ^λ (C)∣≦ ( max {q,λ ₁ n, λ ₁ ′ n}) ^{b ⋅ supp(C)} where supp (C) is the number of non-fixed digits under the action of a permutation in C, λ ₁ is the size of the largest part in λ, and λ ₁ ′ is the number of parts in λ. The proof is obtained by enumeration of rim hook tableaux, the Hook formula and probabilistic arguments. Combinatorial, algebraic and statistical applications follow this result. In particular, we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes. Oblatum 14-III-1995 & 30-X-1995