| 1. |
- Björnberg, Jakob, 1983, et al.
(författare)
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Characterising random partitions by random colouring
- 2020
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 25
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Tidskriftsartikel (refereegranskat)abstract
- Let (X-1, X-2, ...) be a random partition of the unit interval [0, 1], i.e. X-i >= 0 and Sigma(i >= 1) X-i = 1, and let (epsilon(1), epsilon(2), ...) be i.i.d. Bernoulli random variables of parameter p is an element of (0, 1). The Bernoulli convolution of the partition is the random variable Z = Sigma(i >= 1) epsilon X-i(i). The question addressed in this article is: Knowing the distribution of Z for some fixed p is an element of (0, 1), what can we infer about the random partition (X-1, X-2, ...)? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter p is not equal to 1/2.
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| 2. |
- Björnberg, Jakob, 1983, et al.
(författare)
-
Correlation inequalities for the uniform eight-vertex model and the toric code model
- 2023
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Ingår i: Reviews in Mathematical Physics. - 1793-6659 .- 0129-055X. ; 35:10
-
Tidskriftsartikel (refereegranskat)abstract
- We investigate connections between four models in statistical physics and probability theory: (1) the toric code model of Kitaev, (2) the uniform eight-vertex model, (3) random walk on a hypercube, and (4) a classical Ising model with four-body interaction. As a consequence of our analysis (and of the GKS-inequalities for the Ising model) we obtain correlation inequalities for the toric code model and the uniform eight-vertex model.
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| 3. |
- Björnberg, Jakob, 1983, et al.
(författare)
-
Critical parameter of random loop model on trees
- 2018
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Ingår i: Annals of Applied Probability. - 1050-5164. ; 28:4, s. 2063-2082
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Tidskriftsartikel (refereegranskat)abstract
- We give estimates of the critical parameter for random loop models that are related to quantum spin systems. A special case of the model that we consider is the interchange- or random-stirring process. We consider here the model defined on regular trees of large degrees, which are expected to approximate high spatial dimensions. We find a critical parameter that indeed shares similarity with existing numerical results for the cubic lattice. In the case of the interchange process, our results improve on earlier work by Angel and by Hammond, in that we determine the second-order term of the critical parameter.
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| 4. |
- Björnberg, Jakob, 1983, et al.
(författare)
-
Critical Temperature of Heisenberg Models on Regular Trees, via Random Loops
- 2018
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Ingår i: Journal of Statistical Physics. - : Springer Science and Business Media LLC. - 0022-4715 .- 1572-9613. ; 173:5, s. 1369-1385
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Tidskriftsartikel (refereegranskat)abstract
- We estimate the critical temperature of a family of quantum spin systems on regular trees of large degree. The systems include the spin-1/2 XXZ model and the spin-1 nematic model. Our formula is conjectured to be valid for large-dimensional cubic lattices. Our method of proof uses a probabilistic representation in terms of random loops.
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| 5. |
- Björnberg, Jakob, 1983, et al.
(författare)
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Decay of transverse correlations in quantum Heisenberg models
- 2015
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Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 56:4
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Tidskriftsartikel (refereegranskat)abstract
- We study a class of quantum spin systems that include the S = 1/2 Heisenberg and XY-models and prove that two-point correlations exhibit exponential decay in the presence of a transverse magnetic field. The field is not necessarily constant, it may be random, and it points in the same direction. Our proof is entirely probabilistic and it relies on a random loop representations of the correlation functions, on stochastic domination and on first-passage percolation.
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| 6. |
- Björnberg, Jakob, 1983, et al.
(författare)
-
Dimerization in Quantum Spin Chains with O(n) Symmetry
- 2021
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 1432-0916 .- 0010-3616. ; 387:2, s. 1151-1189
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Tidskriftsartikel (refereegranskat)abstract
- We consider quantum spins with S >= 1, and two-body interactions with O (2S + 1) symmetry. We discuss the ground state phase diagram of the one-dimensional system. We give a rigorous proof of dimerization for an open region of the phase diagram, for S sufficiently large. We also prove the existence of a gap for excitations.
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| 7. |
- Björnberg, Jakob E., 1983, et al.
(författare)
-
Characterising random partitions by random colouring
- 2020
-
Ingår i: Electronic Communications in Probability. - : Institute of Mathematical Statistics. - 1083-589X. ; 25
-
Tidskriftsartikel (refereegranskat)abstract
- Let (X-1, X-2, ...) be a random partition of the unit interval [0, 1], i.e. X-i >= 0 and Sigma(i >= 1) X-i = 1, and let (epsilon(1), epsilon(2), ...) be i.i.d. Bernoulli random variables of parameter p is an element of (0, 1). The Bernoulli convolution of the partition is the random variable Z = Sigma(i >= 1) epsilon X-i(i). The question addressed in this article is: Knowing the distribution of Z for some fixed p is an element of (0, 1), what can we infer about the random partition (X-1, X-2, ...)? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter p is not equal to 1/2.
|
|
| 8. |
- Björnberg, Jakob E., 1983, et al.
(författare)
-
Correlation inequalities for the uniform eight-vertex model and the toric code model
- 2023
-
Ingår i: Reviews in Mathematical Physics. - 0129-055X. ; 35:10
-
Tidskriftsartikel (refereegranskat)abstract
- We investigate connections between four models in statistical physics and probability theory: (1) the toric code model of Kitaev, (2) the uniform eight-vertex model, (3) random walk on a hypercube, and (4) a classical Ising model with four-body interaction. As a consequence of our analysis (and of the GKS-inequalities for the Ising model) we obtain correlation inequalities for the toric code model and the uniform eight-vertex model.
|
|
| 9. |
- Björnberg, Jakob E., 1983, et al.
(författare)
-
Critical parameter of random loop model on trees
- 2018
-
Ingår i: Annals of Applied Probability. - : Institute of Mathematical Statistics. - 1050-5164. ; 28:4, s. 2063-2082
-
Tidskriftsartikel (refereegranskat)abstract
- We give estimates of the critical parameter for random loop models that are related to quantum spin systems. A special case of the model that we consider is the interchange- or random-stirring process. We consider here the model defined on regular trees of large degrees, which are expected to approximate high spatial dimensions. We find a critical parameter that indeed shares similarity with existing numerical results for the cubic lattice. In the case of the interchange process, our results improve on earlier work by Angel and by Hammond, in that we determine the second-order term of the critical parameter.
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| 10. |
- Björnberg, Jakob Erik, 1983-
(författare)
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Graphical representations of Ising and Potts models : Stochastic geometry of the quantum Ising model and the space-time Potts model
- 2009
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- HTML clipboard Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models. In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate space–time random-cluster model and explore a range of useful probabilistic techniques for studying it. The space– time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in . In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory— and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which gives the results. In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in and in ‘star-like’ geometries.
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