The Brown measure of the free multiplicative Brownian motion

Driver, Bruce K.; Hall, Brian C.; Kemp, Todd

doi:10.1007/s00440-022-01142-z

Cited by 6 publications

(5 citation statements)

References 18 publications

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“…for all t > 0. This is equivalent to (a generalization to arbitrary starting point of) Lemma 5.2 in [18].…”

Section: Andmentioning

confidence: 99%

“…The result is not stated in exactly this way, but this interpretation is derived in Remark 3.5.4. As an application, we demonstrate in Example 3.5.5 how to use Theorem 3.5.3 to give simple, computationally transparent (re-)proofs of some key identities from [18,24,15,22] that are used in the computation of Brown measures of solutions to various free SDEs.…”

Section: Summary and Guide To Readingmentioning

confidence: 99%

“…Special cases of Equation ( 33) have shown up in the calculation of Brown measures of solutions to various free SDEs. Please see [18,24,15,22]. Thus far, such equations are proven in the literature using power series arguments.…”

Section: The Traced Formulamentioning

confidence: 99%

“…For concreteness, we demonstrate how Equation ( 33) can be used to re-prove a key identity (Lemma 5.2 in [18]) that B.K. Driver, B.C.…”

Section: The Traced Formulamentioning

confidence: 99%

“…P. Biane and R. Speicher developed in [4] a theory of free stochastic calculus with respect to semicircular Brownian motion that has yielded many fruitful applications -e.g., to the theories of free SDEs [6,14,20,28], entropy [3], and transport [11]; analysis on Wigner space [4,29]; and the calculation of Brown measures [18,24,15,22]. In this paper, we present an extension and reinterpretation of this free stochastic calculus that naturally connects the Itôtype formulas thereof to the theory of multiple operator integrals (MOIs) via the class N C k (R) of noncommutative C k functions (Definition 4.1.5) introduced by the author in [34].…”

mentioning

confidence: 99%

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Itô's formula for noncommutative $C^2$ functions of free Itô processes

Nikitopoulos

2022

Doc. Math.

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In a recent paper, the author introduced a rich class N C k (R) of "noncommutative C k " functions R → C whose operator functional calculus is k-times differentiable and has derivatives expressible in terms of multiple operator integrals (MOIs). In the present paper, we explore a connection between free stochastic calculus and the theory of MOIs by proving an Itô formula for noncommutative C 2 functions of self-adjoint free Itô processes. To do this, we first extend P. Biane and R. Speicher's theory of free stochastic calculus -including their free Itô formula for polynomials -to allow free Itô processes driven by multiple freely independent semicircular Brownian motions. Then, in the self-adjoint case, we reinterpret the objects appearing in the free Itô formula for polynomials in terms of MOIs. This allows us to enlarge the class of functions for which one can formulate and prove a free Itô formula from the space originally considered by Biane and Speicher (Fourier transforms of complex measures with two finite moments) to the strictly larger space N C 2 (R). Along the way, we also obtain a useful "traced" Itô formula for arbitrary C 2 scalar functions of self-adjoint free Itô processes. Finally, as motivation, we study an Itô formula for C 2 scalar functions of N × N Hermitian matrix Itô processes.

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“…for all t > 0. This is equivalent to (a generalization to arbitrary starting point of) Lemma 5.2 in [18].…”

Section: Andmentioning

confidence: 99%

Section: Summary and Guide To Readingmentioning

confidence: 99%

Section: The Traced Formulamentioning

confidence: 99%

“…For concreteness, we demonstrate how Equation ( 33) can be used to re-prove a key identity (Lemma 5.2 in [18]) that B.K. Driver, B.C.…”

Section: The Traced Formulamentioning

confidence: 99%

mentioning

confidence: 99%

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Itô's formula for noncommutative $C^2$ functions of free Itô processes

Nikitopoulos

2022

Doc. Math.

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Complexity is not enough for randomness

Guo,

Sasieta,

Swingle

2024

SciPost Phys.

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We study the dynamical generation of randomness in Brownian systems as a function of the degree of locality of the Hamiltonian. We first express the trace distance to a unitary design for these systems in terms of an effective equilibrium thermal partition function, and provide a set of conditions that guarantee a linear time to design. We relate the trace distance to design to spectral properties of the time-evolution operator. We apply these considerations to the Brownian pp-SYK model as a function of the degree of locality pp. We show that the time to design is linear, with a slope proportional to 1/p1/p. We corroborate that when pp is of order the system size this reproduces the behavior of a completely non-local Brownian model of random matrices. For the random matrix model, we reinterpret these results from the point of view of classical Brownian motion in the unitary manifold. Therefore, we find that the generation of randomness typically persists for exponentially long times in the system size, even for systems governed by highly non-local time-dependent Hamiltonians. We conjecture this to be a general property: there is no efficient way to generate approximate Haar random unitaries dynamically, unless a large degree of fine-tuning is present in the ensemble of time-dependent Hamiltonians. We contrast the slow generation of randomness to the growth of quantum complexity of the time-evolution operator. Using known bounds on circuit complexity for unitary designs, we obtain a lower bound determining that complexity grows at least linearly in time for Brownian systems. We argue that these bounds on circuit complexity are far from tight and that complexity grows at a much faster rate, at least for non-local systems.

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Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions

Zhong

2022

J. Eur. Math. Soc.

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Let Z_N be a Ginibre ensemble and let A_N be a Hermitian random matrix independent of Z_N such that A_N converges in distribution to a self-adjoint random variable x_0 in a W^* -probability space (\mathscr{A},\tau) . For each t>0 , the random matrix A_N+\sqrt{t}\,Z_N converges in \ast -distribution to x_0+c_t , where c_t is a circular variable of variance t , freely independent of x_0 . We use the Hamilton–Jacobi method to compute the Brown measure \rho_t of x_0+c_t . The Brown measure has a density that is constant along the vertical direction inside the support. The support of the Brown measure of x_0+c_t is related to the subordination function of the free additive convolution of x_0+s_t , where s_t is a semicircular variable of variance t , freely independent of x_0 . Furthermore, the push-forward of \rho_t by a natural map is the law of x_0+s_t . Let G_N(t) be the Brownian motion on the general linear group and let U_N be a unitary random matrix independent of G_N(t) such that U_N converges in distribution to a unitary random variable u in (\mathscr{A},\tau) . The random matrix U_NG_N(t) converges in \ast -distribution to ub_t where b_t is the free multiplicative Brownian motion, freely independent of u . We compute the Brown measure \mu_t of ub_t , extending the recent work by Driver–Hall–Kemp, which corresponds to the case u=I . The measure has a density of the special form \frac{1}{r^2}w_t(\theta) in polar coordinates in its support. The support of \mu_t is related to the subordination function of the free multiplicative convolution of uu_t where u_t is the free unitary Brownian motion, freely independent of u . The push-forward of \mu_t by a natural map is the law of uu_t . In the special case that u is Haar unitary, the Brown measure \mu_t follows the annulus law . The support of the Brown measure of ub_t is an annulus with inner radius e^{-t/2} and outer radius e^{t/2} . In its support, the density in polar coordinates is given by \frac{1}{2\pi t}\,\frac{1}{r^2}.

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The Brown measure of the free multiplicative Brownian motion

Abstract: The free multiplicative Brownian motion $$b_{t}$$ b t is the large-N limit of the Brownian motion on $$\mathsf {GL}(N;\mathbb {C}),$$ GL ( N ; C ) , in the sense of $$*$$ ∗ … Show more

Cited by 6 publications

References 18 publications

Itô's formula for noncommutative $C^2$ functions of free Itô processes

Itô's formula for noncommutative $C^2$ functions of free Itô processes

Complexity is not enough for randomness

Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions

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