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Reflection positivity and phase transitions in lattice fermion systems
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Reflection positivity plays an important role in establishing phase transitions in lattice models with continuous symmetries. Recent developments extend these techniques to some lattice fermion systems. In this talk, I will present recent results for lattice fermion models that have reflection positivity and exhibit long-range order. The talk is based on joint work with Tohru Koma.
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Weak coupling limit of the Pauli-Fierz model
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We investigate the weak coupling limit of the Pauli-Fierz Hamiltonian within a mathematically rigorous framework. Furthermore, we establish the asymptotic behavior of the effective mass in this regime.
Reference:
https://arxiv.org/abs/2505.07253
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Lace expansion for the quantum Ising model
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The lace expansion is a powerful tool for analyzing critical phenomena. It has been successful in establishing mean-field behavior for a variety of models in statistical mechanics. One of our objectives is to obtain a lace expansion for the quantum (transverse-field) Ising model, a model of ferromagnetism. It is defined by applying a transverse field to the classical Ising model. The transverse field yields the non-commutativity and makes the system occur a different type of phase transition from the classical case.
In this talk, I will show the derivation of a lace expansion for the quantum Ising model and upper bounds on the expansion coefficients. The derivation is based on the random current representation [Bj\"{o}rnberg and Grimmett (2009) \textit{J. Stat. Phys.}] [Crawford and Ioffe (2010) \textit{Commun. Math. Phys.}] on space-time. This representation provides the connectivity between two vertices in space-time, which is similar to percolation models and helps us to derive the lace expansion. Also, we obtain upper bounds on the expansion coefficients by separating a connected component into some events. For example, a double connection is separated into connections shaped like ``laces''.
This talk is based on joint work with Akira Sakai (Hokkaido University, Japan).
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Symmetry-protected topological phase from the viewpoint of homotopy theory
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An SPT (symmetry-protected topological) phase is a kind of topological phase of matter that only arises when one takes into account the on-site symmetry of a group. A typical example is the AKLT phase, which is distinguished with the trivial phase by a topological invariant taking values in $\mathbb{Z}/2$. For quantum spin systems in dimensions $\leq 2$, it has been known that there exist topological invariants taking values in the group cohomology of $G$. As a conceptual framework to understand these invariants, Kitaev proposed the statement that “the space of invertible gapped ground states forms an $\Omega$-spectrum.” In this talk, I will present the speaker’s result which provides a mathematical formulation and proof of this conjecture.
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Subcritical two-point functions for high-dimensional statistical mechanical models
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The subcritical two-point function for the Ising model, percolation, or the self-avoiding walk on $\mathbb{Z}^d$, decays like its critical counterpart when $x$ is small, and exhibits the Ornstein--Zernike decay when $x$ is large. We report recent progress on the slightly subcritical two-point function for the high-dimensional self-avoiding walk, where we obtained asymptotics that are uniform in the distance to criticality. In particular, our result reduces to the two regimes of decay, and we identify the order of the constant in the Ornstein--Zernike decay. This is based on an ongoing work with Gordon Slade.
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Multiplicative Averages of Plancherel Random Partitions: Elliptic Functions, Phase Transitions, and Applications
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We consider random integer partitions λ distributed according to the Poissonized Plancherel measure with parameter t². Using Riemann–Hilbert techniques, we establish the asymptotics of certain multiplicative averages of the form
Q(t, s) = E [ ∏_{i ≥ 1} (1 + q^(s + i – λ_i – 1/2))^(-1) ],
for fixed q in the interval (0,1), in the regime where both s and t are large. We compute the large-t expansion
log Q(t, x t) = –t² F_q(x) + O(1),
expressing the rate function F_q(x) explicitly in terms of Weierstrass elliptic functions. The associated equilibrium measure generally exhibits saturated regions and undergoes two distinct third-order phase transitions, which we describe in detail. Applications of our results include explicit characterizations of the tail probabilities of the height function in the q-deformed Polynuclear Growth model, of the edge behavior of the positive-temperature discrete Bessel process, and the asymptotics of the cylindrical Toda chain with shock-type initial data.
Based on a joint work with Mattia Cafasso and Giulio Ruzza.
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One-arm exponents of the high-dimensional Ising model
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We study the probability that the origin is connected to the boundary of the box of size n in several percolation models related to the Ising model. Of particular interest to us is the random cluster representation of the Ising model (or FK-Ising model). We prove that different universality classes emerge at criticality depending on the choice of boundary condition. Based on a joint work with Diederik van Engelenburg, Christophe Garban, and Franco Severo.
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Finite-size scalings of the Euclidean phi4 model at and above the critical dimension
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The study of finite-size scaling in systems at and above their upper critical dimension, particularly concerning the breaking of the hyperscaling relation. For models in the Ising universality class Renormalisation Group (RG) arguments—such as those by Brézin and Zinn-Justin (1978) on the susceptibility profile and the work by Flores-Sola, Berche, Kenna and Weigel (2016) on the role of the other Fourier models—provided key predictions for this behaviour. In my talk, I will discuss theorems that verify these predictions based on a rigorous RG argument.
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Conservation operator processes and their CLT from asymptotic representation theory
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In this talk, we examine applications of the theory of operator-valued processes to algebraic methods in probability theory. In particular, we analyze the asymptotic behavior of conservation operator processes, which are defined on a symmetric Fock space, derived from unitary groups and quantum unitary groups as their rank tends to infinity.
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A random walk approach to high-dimensional critical phenomena
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The study of critical phenomena in lattice statistical mechanical models such as percolation and spin systems has a long history in both physics and mathematics. A central problem is to prove existence and calculate the values of the critical exponents that govern the universal behaviour of the model at and near its critical point. Detailed proofs are typically available only for models in dimension two, or above an upper critical dimension where mean-field behaviour is observed. We present a new, unified, generic, probabilistic, and relatively very simple proof of mean-field critical behaviour for high-dimensional models containing a small parameter. The results apply to spin systems and self-avoiding walk in dimensions above 4, percolation in dimensions above 6, and lattice trees in dimensions above 8. Minimal model-specific data is required for these applications. This is joint work with Hugo Duminil-Copin, Aman Markar and Romain Panis.
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Microscopic and Macroscopic Hall-Current Response in Infinitely-Extended Systems
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The bulk of an experimental quantum Hall setup can be mathematically modeled as an infinitely extended two-dimensional lattice system at zero temperature with a spectral gap. If one applies an external electric potential to such a system, a current perpendicular to the potential gradient is induced. This induced current is called the Hall-current response.
In this talk I will explain how the Hall-conductance and Hall-conductivity arise as distinct linear-response coefficients, associated to microscopic and macroscopic Hall-current responses, respectively. I will discuss how one can rigorously show the existence of these coefficients via the NEASS (non-equilibrium almost-stationary state) approach to linear-response theory, which results in the much stronger statement that the Hall-current response is nearly linear in the strength of the potential with no polynomial corrections, giving a justification of Ohm's law.
This talk is based on a series of joint works with Giovanna Marcelli, Tadahiro Miyao, Domenico Monaco, and Stefan Teufel.