Browsing by Author "Its, Alexander R."
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Item Asymptotics of the Fredholm determinant corresponding to the first bulk critical universality class in random matrix models(2013-11-06) Bothner, Thomas Joachim; Its, Alexander R.; Bleher, Pavel, 1947-; Tarasov, Vitaly; Eremenko, Alexandre; Mukhin, EvgenyWe study the one-parameter family of determinants $det(I-\gamma K_{PII}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator $K_{PII}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\'e equation. In case $\gamma=1$, this Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-\gamma K_{PII})$ for all values of the real parameter $\gamma$.Item Exact Solutions to the Six-Vertex Model with Domain Wall Boundary Conditions and Uniform Asymptotics of Discrete Orthogonal Polynomials on an Infinite Lattice(2011-03-09) Liechty, Karl Edmund; Bleher, Pavel, 1947-; Its, Alexander R.; Lempert, Lazlo; Kitchens, Bruce, 1953-In this dissertation the partition function, $Z_n$, for the six-vertex model with domain wall boundary conditions is solved in the thermodynamic limit in various regions of the phase diagram. In the ferroelectric phase region, we show that $Z_n=CG^nF^{n^2}(1+O(e^{-n^{1-\ep}}))$ for any $\ep>0$, and we give explicit formulae for the numbers $C, G$, and $F$. On the critical line separating the ferroelectric and disordered phase regions, we show that $Z_n=Cn^{1/4}G^{\sqrt{n}}F^{n^2}(1+O(n^{-1/2}))$, and we give explicit formulae for the numbers $G$ and $F$. In this phase region, the value of the constant $C$ is unknown. In the antiferroelectric phase region, we show that $Z_n=C\th_4(n\om)F^{n^2}(1+O(n^{-1}))$, where $\th_4$ is Jacobi's theta function, and explicit formulae are given for the numbers $\om$ and $F$. The value of the constant $C$ is unknown in this phase region. In each case, the proof is based on reformulating $Z_n$ as the eigenvalue partition function for a random matrix ensemble (as observed by Paul Zinn-Justin), and evaluation of large $n$ asymptotics for a corresponding system of orthogonal polynomials. To deal with this problem in the antiferroelectric phase region, we consequently develop an asymptotic analysis, based on a Riemann-Hilbert approach, for orthogonal polynomials on an infinite regular lattice with respect to varying exponential weights. The general method and results of this analysis are given in Chapter 5 of this dissertation.Item Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge(Springer, 2016-10) Bogatskiy, A.; Claeys, T.; Its, Alexander R.; Department of Mathematical Sciences, School of ScienceWe compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz–Segur family of solutions to the Painlevé II equation. Our results complement the ones in [33]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlevé II transcendents, and we also prove a new result on the poles of the Ablowitz–Segur solutions to the Painlevé II equation. We also highlight applications of our results in random matrix theory.Item Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa I. Stokes Data(Oxford, 2015-02) Guest, Martin A.; Its, Alexander R.; Lin, Chang-Shou; Department of Mathematical Sciences, School of ScienceWe describe all smooth solutions of the two-function tt*-Toda equations (a version of the tt* equations or equations for harmonic maps into SLnR/SOn SLnR/SOn ) in terms of (1) asymptotic data, (2) holomorphic data, and (3) monodromy data, and we compute all of this data explicitly. This allows us, in particular, to find all solutions with integral Stokes data. These include solutions associated to non-linear sigma models (quantum cohomology) or Landau–Ginzburg models (unfoldings of singularities), as conjectured by Cecotti and Vafa in the 1990s.Item Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa III: Iwasawa Factorization and Asymptotics(Springer, 2020-03) Guest, Martin A.; Its, Alexander R.; Lin, Chang-Shou; Mathematical Sciences, School of ScienceThis paper, the third in a series, completes our description of all (radial) solutions on C∗ of the tt*-Toda equations 2(wi)tt¯=−e2(wi+1−wi)+e2(wi−wi−1), using a combination of methods from p.d.e., isomonodromic deformations (Riemann–Hilbert method), and loop groups. We place these global solutions into the broader context of solutions which are smooth near 0. For such solutions, we compute explicitly the Stokes data and connection matrix of the associated meromorphic system, in the resonant cases as well as the non-resonant case. This allows us to give a complete picture of the monodromy data, holomorphic data, and asymptotic data of the global solutions.Item Large-x analysis of an operator valued Riemann–Hilbert problem(Oxford, 2016) Its, Alexander R.; Kozlowski, K. K.; Department of Mathematical Sciences, School of ScienceThe purpose of this paper is to push forward the theory of operator-valued Riemann–Hilbert problems and demonstrate their effectiveness in respect to the implementation of a non-linear steepest descent method á la Deift–Zhou. In this paper, we demonstrate that the operator-valued Riemann–Hilbert problem arising in the characterization of so-called cc-shifted integrable integral operators allows one to extract the large-xx asymptotics of the Fredholm determinant associated with such operators.Item Mathematical Models of Basal Ganglia Dynamics(2013-07-12) Dovzhenok, Andrey A.; Rubchinsky, Leonid; Kuznetsov, Alexey; Its, Alexander R.; Worth, Robert; Mukhin, EvgenyPhysical and biological phenomena that involve oscillations on multiple time scales attract attention of mathematicians because resulting equations include a small parameter that allows for decomposing a three- or higher-dimensional dynamical system into fast/slow subsystems of lower dimensionality and analyzing them independently using geometric singular perturbation theory and other techniques. However, in most life sciences applications observed dynamics is extremely complex, no small parameter exists and this approach fails. Nevertheless, it is still desirable to gain insight into behavior of these mathematical models using the only viable alternative – ad hoc computational analysis. Current dissertation is devoted to this latter approach. Neural networks in the region of the brain called basal ganglia (BG) are capable of producing rich activity patterns. For example, burst firing, i.e. a train of action potentials followed by a period of quiescence in neurons of the subthalamic nucleus (STN) in BG was shown to be related to involuntary shaking of limbs in Parkinson’s disease called tremor. The origin of tremor remains unknown; however, a few hypotheses of tremor-generation were proposed recently. The first project of this dissertation examines the BG-thalamo-cortical loop hypothesis for tremor generation by building physiologically-relevant mathematical model of tremor-related circuits with negative delayed feedback. The dynamics of the model is explored under variation of connection strength and delay parameters in the feedback loop using computational methods and data analysis techniques. The model is shown to qualitatively reproduce the transition from irregular physiological activity to pathological synchronous dynamics with varying parameters that are affected in Parkinson’s disease. Thus, the proposed model provides an explanation for the basal ganglia-thalamo-cortical loop mechanism of tremor generation. Besides tremor-related bursting activity BG structures in Parkinson’s disease also show increased synchronized activity in the beta-band (10-30Hz) that ultimately causes other parkinsonian symptoms like slowness of movement, rigidity etc. Suppression of excessively synchronous beta-band oscillatory activity is believed to suppress hypokinetic motor symptoms in Parkinson’s disease. Recently, a lot of interest has been devoted to desynchronizing delayed feedback deep brain stimulation (DBS). This type of synchrony control was shown to destabilize synchronized state in networks of simple model oscillators as well as in networks of coupled model neurons. However, the dynamics of the neural activity in Parkinson’s disease exhibits complex intermittent synchronous patterns, far from the idealized synchronized dynamics used to study the delayed feedback stimulation. The second project of this dissertation explores the action of delayed feedback stimulation on partially synchronous oscillatory dynamics, similar to what one observes experimentally in parkinsonian patients. We employ a computational model of the basal ganglia networks which reproduces the fine temporal structure of the synchronous dynamics observed experimentally. Modeling results suggest that delayed feedback DBS in Parkinson’s disease may boost rather than suppresses synchronization and is therefore unlikely to be clinically successful. Single neuron dynamics may also have important physiological meaning. For instance, bistability – coexistence of two stable solutions observed experimentally in many neurons is thought to be involved in some short-term memory tasks. Bistability that occurs at the depolarization block, i.e. a silent depolarized state a neuron enters with excessive excitatory input was proposed to play a role in improving robustness of oscillations in pacemaker-type neurons. The third project of this dissertation studies what parameters control bistability at the depolarization block in the three-dimensional conductance-based neuronal model by comparing the reduced dopaminergic neuron model to the Hodgkin-Huxley model of the squid giant axon. Bifurcation analysis and parameter variations revealed that bistability is mainly characterized by the inactivation of the Na+ current, while the activation characteristics of the Na+ and the delayed rectifier K+ currents do not account for the difference in bistability in the two models.Item Monodromy dependence and connection formulae for isomonodromic tau functions(Duke, 2018) Its, Alexander R.; Lisovyy, O.; Prokhorov, Andrei; Mathematical Sciences, School of ScienceWe discuss an extension of the Jimbo–Miwa–Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola, generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for the generic Painlevé VI tau function. The result proves the conjectural formula for this constant proposed by Iorgov, Lisovyy, and Tykhyy. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of the Painlevé II tau function.Item On Some Hamiltonian Properties of the Isomonodromic Tau Functions(World Scientific, 2018-08) Its, Alexander R.; Prokhorov, A.; Mathematical Sciences, School of ScienceWe discuss some new aspects of the theory of the Jimbo–Miwa–Ueno tau function which have come to light within the recent developments in the global asymptotic analysis of the tau functions related to the Painlevé equations. Specifically, we show that up to the total differentials the logarithmic derivatives of the Painlevé tau functions coincide with the corresponding classical action differential. This fact simplifies considerably the evaluation of the constant factors in the asymptotics of tau functions, which has been a long-standing problem of the asymptotic theory of Painlevé equations. Furthermore, we believe that this observation is yet another manifestation of L. D. Faddeev’s emphasis of the key role which the Hamiltonian aspects play in the theory of integrable system.Item Riemann–Hilbert approach to a generalized sine kernel(Springer Link, 2020-02-01) Gharakhloo, Roozbeh; Its, Alexander R.; Kozlowski, Karol K.; Mathematical Sciences, School of ScienceWe derive the large-distance asymptotics of the Fredholm determinant of the so-called generalized sine kernel at the critical point. This kernel corresponds to a generalization of the pure sine kernel arising in the theory of random matrices and has potential applications to the analysis of the large-distance asymptotic behaviour of the so-called emptiness formation probability for various quantum integrable models away from their free fermion point.