Normalized Partition Function Research Articles

A.s. convergence rate for Mandelbrot’s cascade in a random environment

. For Mandelbrot’s cascade (Y n ) in an independent and identically distributed (i.i.d.) random environment ξ, we are interested in the a.s. convergence rate of the Mandelbrot’s martingale (W n ) to its limit W, where W n = Y n / E ξ Y n is the normalized partition function. We obtain sufficient conditions under which W−W n has an exponential convergence rate: W − W n = o ( e − n a ) a.s. for some a > 0 explicitly calculated; we also find conditions under which W−W n has a polynomial convergence rate: W − W n = o ( n − α ) a.s. for some α > 0. Similar conclusions hold for Mandelbrot’s cascade in a varying environment.

Multifractal Company Market: An Application to the Stock Market Indices.

Using the multiscale normalized partition function, we exploit the multifractal analysis based on directly measurable shares of companies in the market. We present evidence that markets of competing firms are multifractal/multiscale. We verified this by (i) using our model that described the critical properties of the company market and (ii) analyzing a real company market defined by the index. As the valuable reference case, we considered a four-group market model that skillfully reconstructs this index’s empirical data. We point out that a four-group company market organization is universal because it can perfectly describe the essential features of the spectrum of dimensions, regardless of the analyzed series of shares. The apparent differences from the empirical data appear only at the level of subtle effects.

Gaussian fluctuations for the directed polymer partition function in dimension d≥3 and in the whole L2-region

We consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations of the tail n(d−2)/4(W∞−Wn) of the normalized partition function. It was proven by Comets and Liu (J. Math. Anal. Appl. 455 (2017) 312–335), that for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting partition function and an independent Gaussian random variable. We extend the result to the whole L2-region of temperature, which is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the considered scaling. To do so, we manage to avoid the heavy 4th-moment computation and instead rely on the local limit theorem for polymers (Fund. Math. 147 (1995) 173–180; Ann. Inst. Henri Poincare Probab. Stat. 42 (2006) 521–534) and homogenization.

1-stable fluctuations in branching Brownian motion at critical temperature I: The derivative martingale

Let $(Z_{t})_{t\geq 0}$ denote the derivative martingale of branching Brownian motion, that is, the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke (Ann. Probab. 15 (1987) 1052–1061) says that this martingale converges almost surely to a limit $Z_{\infty }$, positive on the event of survival. In this paper our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture by Mueller and Munier (Phys. Rev. E 90 (2014) 042143): \begin{equation*}\sqrt{t}\bigg(Z_{\infty }-Z_{t}+\frac{\log t}{\sqrt{2\pi t}}Z_{\infty }\bigg)\xrightarrow[t\to \infty ]{}S_{Z_{\infty }}\quad\text{in law},\end{equation*} where $S$ is a spectrally positive 1-stable Lévy process independent of $Z_{\infty }$. In a first part of the paper, a relatively short proof of (a slightly stronger form of) this convergence is given based on the functional equation satisfied by the characteristic function of $Z_{\infty }$ together with tail asymptotics of this random variable. We then set up more elaborate arguments which yield a more thorough understanding of the trajectories of the particles contributing to the fluctuations. In this way we can upgrade our convergence result to functional convergence. This approach also sets the ground for a follow-up paper, where we study the fluctuations of more general functionals including the renormalized critical additive martingale. All proofs in this paper are given under the moment assumption $\mathbb{E}[L(\log L)^{3}]<\infty $, where the random variable $L$ follows the offspring distribution of the branching Brownian motion. We believe this hypothesis to be optimal.

BPS/CFT Correspondence III: Gauge Origami Partition Function and qq-Characters

We study generalized gauge theories engineered by taking the low energy limit of the $Dp$ branes wrapping $X \times T^{p-3}$, with $X$ a possibly singular surface in a Calabi-Yau fourfold $Z$. For toric $Z$ and $X$ the partition function can be computed by localization, making it a statistical mechanical model, called the gauge origami. The random variables are the ensembles of Young diagrams. The building block of the gauge origami is associated with a tetrahedron, whose edges are colored by vector spaces. We show the properly normalized partition function is an entire function of the Coulomb moduli, for generic values of the $\Omega$-background parameters. The orbifold version of the theory defines the $qq$-character operators, with and without the surface defects. The analytic properties are the consequence of a relative compactness of the moduli spaces $M({\vec n}, k)$ of crossed and spiked instantons, demonstrated in arXiv:1608.07272.

Nested Critical Points for a Directed Polymer on a Disordered Diamond Lattice

We consider a model for a directed polymer in a random environment defined on a hierarchical diamond lattice in which i.i.d. random variables are attached to the lattice bonds. Our focus is on scaling schemes in which a size parameter n, counting the number of hierarchical layers of the system, becomes large as the inverse temperature $$\beta $$ vanishes. When $$\beta $$ has the form $${\widehat{\beta }}/\sqrt{n}$$ for a parameter $${\widehat{\beta }}>0$$ , we show that there is a cutoff value $$0< \kappa < \infty $$ such that as $$n \rightarrow \infty $$ the variance of the normalized partition function tends to zero for $${\widehat{\beta }}\le \kappa $$ and grows without bound for $${\widehat{\beta }}> \kappa $$ . We obtain a more refined description of the border between these two regimes by setting the inverse temperature to $$\kappa /\sqrt{n} + \alpha _n$$ where $$0 < \alpha _n \ll 1/\sqrt{n}$$ and analyzing the asymptotic behavior of the variance. We show that when $$\alpha _n = \alpha (\log n-\log \log n)/n^{3/2}$$ (with a small modification to deal with non-zero third moment), there is a similar cutoff value $$\eta $$ for the parameter $$\alpha $$ such that the variance goes to zero when $$\alpha < \eta $$ and grows without bound when $$\alpha > \eta $$ . Extending the analysis yet again by probing around the inverse temperature $$(\kappa / \sqrt{n}) + \eta (\log n-\log \log n)/n^{3/2}$$ , we find an infinite sequence of nested critical points for the variance behavior of the normalized partition function. In the subcritical cases $${\widehat{\beta }}\le \kappa $$ and $$\alpha \le \eta $$ , this analysis is extended to a central limit theorem result for the fluctuations of the normalized partition function.

Rate of convergence for polymers in a weak disorder

We consider directed polymers in random environment on the lattice Zd at small inverse temperature and dimension d≥3. Then, the normalized partition function Wn is a regular martingale with limit W. We prove that n(d−2)/4(Wn−W)/Wn converges in distribution to a Gaussian law. Both the polynomial rate of convergence and the scaling with the martingale Wn are different from those for polymers on trees.

The intermediate disorder regime for a directed polymer model on a hierarchical lattice

We study a directed polymer model defined on a hierarchical diamond lattice, where the lattice is constructed recursively through a recipe depending on a branching number b∈N and a segment number s∈N. When b≤s it is known that the model exhibits strong disorder for all positive values of the inverse temperature β, and thus weak disorder reigns only for β=0 (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature β≡βn vanishes at an appropriate rate as the size n of the system grows. Our analysis requires separate treatment for the cases b<s and b=s. In the case b<s we prove that when the inverse temperature is taken to be of the form βn=β̂(b/s)n/2 for β̂>0, the normalized partition function of the system converges weakly as n→∞ to a distribution L(β̂) and does so universally with respect to the initial weight distribution. We prove the convergence using renormalization group type ideas rather than the standard Wiener chaos analysis. In the case b=s we find a critical point in the behavior of the model when the inverse temperature is scaled as βn=β̂/n; for an explicitly computable critical value κb>0 the variance of the normalized partition function converges to zero with large n when β̂≤κb and grows without bound when β̂>κb. Finally, we prove a central limit theorem for the normalized partition function when β̂≤κb.

On the Widom–Rowlinson Occupancy Fraction in Regular Graphs

We consider the Widom–Rowlinson model of two types of interacting particles on d-regular graphs. We prove a tight upper bound on the occupancy fraction, the expected fraction of vertices occupied by a particle under a random configuration from the model. The upper bound is achieved uniquely by unions of complete graphs on d + 1 vertices, Kd+1. As a corollary we find that Kd+1 also maximizes the normalized partition function of the Widom–Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalized number of homomorphisms from any d-regular graph G to the graph HWR, a path on three vertices with a loop on each vertex, is maximized by Kd+1. This proves a conjecture of Galvin.

Path-integral invariants in abelian Chern–Simons theory

We consider the U(1) Chern–Simons gauge theory defined in a general closed oriented 3-manifold M; the functional integration is used to compute the normalized partition function and the expectation values of the link holonomies. The non-perturbative path-integral is defined in the space of the gauge orbits of the connections which belong to the various inequivalent U(1) principal bundles over M; the different sectors of configuration space are labelled by the elements of the first homology group of M and are characterized by appropriate background connections. The gauge orbits of flat connections, whose classification is also based on the homology group, control the non-perturbative contributions to the mean values. The functional integration is carried out in any 3-manifold M, and the corresponding path-integral invariants turn out to be strictly related with the abelian Reshetikhin–Turaev surgery invariants.

Three-manifold invariant from functional integration

We give a precise definition and produce a path-integral computation of the normalized partition function of the Abelian U(1) Chern-Simons field theory defined in a general closed oriented 3-manifold. We use the Deligne-Beilinson formalism, we sum over the inequivalent U(1) principal bundles over the manifold and, for each bundle, we integrate over the gauge orbits of the associated connection 1-forms. The result of the functional integration is compared with the Abelian U(1) Reshetikhin-Turaev surgery invariant.

A local limit theorem for directed polymers in random media: the continuous and the discrete case

In this article, we consider two models of directed polymers in random environment: a discrete model in a general random environment and a continuous model. We consider these models in dimension greater than or equal to 3 and we suppose that the normalized partition function is bounded in L 2 (the “high” temperature case). Under these assumptions, Sinai proved in [Y. Sinai, A remark concerning random walks with random potentials, Fund. Math. 147 (1995) 173–180] a local limit theorem for the discrete model, using a perturbation expansion. In this article, we give a new method for proving Sinai's local limit theorem. This new method can be transposed to the continuous setting in which we prove a similar local limit theorem.

Strong Disorder for a Certain Class of Directed Polymers in a Random Environment

We study a model of directed polymers in a random environment with a positive recurrent Markov chain, taking values in a countable space Σ. The random environment is a family ( $$g(i,x), i \geq 1,x \in \Sigma$$ ) of independent and identically distributed real-valued variables. The asymptotic behaviour of the normalized partition function is characterized: when the common law of the g(·,·) is infinitely divisible and the Markov chain is exponentially recurrent we prove that the normalized partition function converges exponentially fast towards zero at all temperatures.

Brownian Directed Polymers in Random Environment

We study the thermodynamics of a continuous model of directed polymers in random environment. The environment is given by a space-time Poisson point process, whereas the polymer is defined in terms of the Brownian motion. We mainly discuss: (i) The normalized partition function, its positivity in the limit which characterizes the phase diagram of the model. (ii) The existence of quenched Lyapunov exponent, its positivity, and its agreement with the annealed Lyapunov exponent; (iii) The longitudinal fluctuation of the free energy, some of its relations with the overlap between replicas and with the transversal fluctuation of the path. The model considered here, enables us to use stochastic calculus, with respect to both Brownian motion and Poisson process, leading to handy formulas for fluctuations analysis and qualitative properties of the phase diagram. We also relate our model to some formulation of the Kardar-Parisi-Zhang equation, more precisely, the stochastic heat equation. Our fluctuation results are interpreted as bounds on various exponents and provide a circumstantial evidence of super-diffusivity in dimension one. We also obtain an almost sure large deviation principle for the polymer measure.

Efficient thermal rate constant calculation for rare event systems

We present an efficient method for computing thermal reaction rate constants that can be applied to systems in which transitions from reactant to product are infrequent. The method can be applied to high-dimensional, disordered systems which exhibit too many transition states to be identified, and for which useful reaction coordinates cannot be easily defined. The focus of our method is the time correlation function C(t), the normalized partition function for trajectories that begin in the reactant region and end in the product region after a time t; the time derivative of C(t) is the reaction rate constant, k(t). We use an umbrella potential to select initial positions from improbable regions of the reactant configuration space. We then compute C(t) directly by choosing random thermal momenta and asking if the resulting dynamical trajectory reaches the product region in time t. Since dynamical trajectories are run on the true potential energy surface, without the umbrella, re-crossing effects are included correctly. The initial condition bias introduced by the umbrella is removed by a weighting factor. We test the method on a simple two dimensional model potential and on a model for the isomerization of a diatomic in a Weeks–Chandler–Andersen fluid, and show that it gives accurate and precise rates with substantial reduction in computer time.

Phase equilibria of a nematic and smectic-A mixture

A phase diagram of a mixture consisting of nematic and smectic liquid crystals has been calculated self-consistently by combining Flory–Huggins (FH) theory for isotropic mixing and Maier–Saupe–McMillan (MSM) theory for smectic-A ordering. However, the MSM theory can be deduced to the original Maier–Saupe (MS) theory for nematic ordering. To describe the phase transitions involving induced smectic phase and nematic + smectic equilibrium, two nematic and two smectic order parameters for the nematic/smectic mixtures have been coupled through the normalized partition function and the orientation distribution function. Self-consistent numerical solution has been sought in establishing nematic/smectic phase diagrams involving (i) phase separation between nematic and smectic liquid crystals and (ii) occurrence of induced smectic in a nematic/smectic mixture. The predictive capability of this combined FH/MSM theory has been tested critically with a reported phase diagram of a nematic/smectic liquid-crystal mixture and also with our experimental phase diagram of a mixture consisting of a nematic side-on side-chain liquid-crystalline polymer and a smectic low molar mass liquid crystal.

Statistical thermodynamic analysis of the excess heat capacity function of macromolecular systems

The deconvolution theory of thermal transitions has proven to be a powerful method with which to analyze the heat capacity function of macromolecular systems. In this article, the basic results of the theory will be presented and their application to multistate transitions and general cooperative transitions of biopolymers and phospholipid membranes will be discussed. INTRODUCTION The development of highly precise differential scanning calorimeters has made possible the accurate definition of the heat capacity function associated with thermally-induced transitions of proteins, polypeptides, nucleic acids, lipid bilayers and other macromolecular systems. The importance of having experimental access to the exact shape of this function is that it contains all the information necessary to develop a complete thermodynamic description of a thermally-induced transition. In fact, it has been demonstrated that the excess heat capacity function can be appropriately transformed to yield the partition function of such a system and that this partition function can be used to deduce the microscopic mechanism of the transition (Refs. 1-6). In this article the analytical methods directed to obtaining a detailed statistical thermodynamic description of complex macromolecular systems will be presented. These methods constitute the basis of the deconvolution theory of thermal-transitions in macromolecules and, thus f.ar, have been applied to the study of protein unfolding reactions, helix-coil transitions in polynucleotides, thermal-transitions of transfer ribonucleic acids (tRNA) and biomembrane phase transitions. MACROMOLECULAR CONFORMATIONAL TRANSITIONS Most theories directed toward describing the molecular basis of function and modulation of biochemical systems include structural variations of the relevant macromolecules. Thus important questions relate to what are the characteristics of the equilibrium and dynamic fluctuations within an ensemble of such states. In general, the accessible structural states of a macromolecular system can be represented by the following reaction scheme: A+A A A (1) 0+ 1+ 2 n where the indexing of states is such that the enthalpy of state i (H.) is greater than the i-l state. (i.e. H>H.,). Thus as temperature i increased the population distribution mohotonically progresses toward state n. A normalized partition function for this system can be written in terms of the Gibbs energy differences as: