Citation 数理解析研究所講究録 (2012), 1805:

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Citation 数理解析研究所講究録 (2012), 1805: 25-36 Issue Date 2012-08 URL http://hdl.

1 Title Ground States for 2$D$ Spin Renormalization Group Methods Glasses in Ma Author(s) Arguin, Louis-Pierre; Damron, Micha Stein, Daniel L. Citation 数理解析研究所講究録 (2012), 1805: Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 Ground States for $2D$ Spin Glasses Louis-Pierre Arguin Universite de Montr\ eal, Canada Michael Damron Princeton University, USA Charles M. Newman New York University, USA Daniel L. Stein New York University, USA Research supported in part by NSF grants DMS , OISE and DMS and an NSF postdoctoral fellowship to M. Damron.

3 26 Edwards-Anderson ( $EA$ ) Spin Glass For $G_{N}=(V_{N}, E_{N})$, graph with $N$ vertices, let $H_{N}( \sigma)=-\sum_{(x,y)\in E_{N}}J_{xy}\sigma_{x}\sigma_{y}$ with $J:=\{J_{xy}\}$ indep. mean zero Gaussian (or other). For How How $SK$ model [SK], is the complete graph. $G_{N}$ For $EA$ $G_{N}\subset \mathbb{z}^{d}.$ model [EA], Spin glasses exhibit many, very different, states of lo energy that lead to complex behavior. many ground states as $Narrow\infty$? different is $EA$ than $SK$? For what? $d$

4 $\triangleright$ Take $\triangleright$ Write $\triangleright$ $\triangleright$ 27 Gibbs States and Ground States $\mu$ The Gibbs measure is concentrated on $\sigma$ s with small $H_{N}.$ $\mathcal{g}_{\beta,n}(\sigma)=\frac{e\cross p-\beta H_{N}(\sigma)}{Z_{N}(\beta)}$ $\mathbb{z}^{d}$ gives information on $\sigma$ s close to the minimum. $\triangleright \mathcal{g}_{\beta,n}$ Take $G_{N}=(V_{N}, E_{N})$, a box in (of volume $N$ ). periodic b.c. $(so H_{N}(\sigma)=H_{N}(-\sigma))$. $:=\{\sigma, -\sigma\}$, a Ground State Pair (GSP). $\alpha$ Study the measure on the unique GSP $\alpha_{n}(j):\delta_{\alpha_{n}(j)}$ Equivalent to studying Gibbs state with $\betaarrow\infty$, then $Narrow\infty$

5 28 Metastate [AW, NSl]: a measure on GSP s $\nu\alpha_{n,j}$ could change drastically between $N$ and $N.$ Consider the joint distribution of $(\alpha_{n,j}, J)$, $K_{N}:=\delta_{\alpha_{N,J}}v_{N}(dJ)$ Take Express (subsequence) limit of as $a$ $K_{N}$ $Narrow\infty.$ the limit $K$ as $K_{J}v(dJ)$. $\nu K_{J}v(dJ)$ is translation invariant. (Periodic b.c.) $\nu K_{J}$ is a measure on GSP s on $\mathbb{z}^{d}$ for given $J.$

6 29 Ground States in Infinite Volume Definition (Ground State Property) $\alpha=\{\sigma, -\sigma\}$ is a $GSP$ on $Z^{d}$ for $J$ if for any finite $A\subset \mathbb{z}^{d}$ $\sum_{(x,y)\in\partial A}J_{xy}\sigma_{x}\sigma_{y}<0$

7 30 Ground States in Low Dimension Conjecture (Uniqueness of Ground States) For low $d(d<d_{c}=6?, =8?, =\infty?)$, the limit $K=K_{J}v(dJ)$ exists (no subseq. needed); $\nu K_{J}$ is supported on a single $GSP.$ $(d=2$ numerics $of Palassini-$Young $[PY],$ Middleton $[M])$ Strategy: $\alpha$ 1. Let and be replica GSP s. 2. Study the interface: $\alpha $ $\alpha\delta\alpha :=\{(x, y):\alpha_{xy}\neq\alpha_{xy} \},$ where $\alpha_{xy}$ 3. Show $\alpha\delta\alpha $ is $:=\sigma_{x}\sigma_{y}.$ empty (hence $\alpha=\alpha $ ).

8 $\triangleright$ 31 Interfaces between GSP s $\triangleright\alpha\delta\alpha =\{(x, y):\alpha_{xy}\neq\alpha_{xy} \}.$ Put a dual edge whenever $(x, y)\in\alpha\triangle\alpha$

9 32 Full Plane Partial Result $\alpha$ If and from $\alpha $ $\nu\alpha\triangle\alpha $ metastate are distinct; then cannot have dangling ends (or 3-branching points). cannot contain loops. cannot have 4-branching points. So $\alpha\delta\alpha $ is one or more doubly-infinite (self-avoiding) paths. Theorem (Newman, Stein [NS2]) $\alpha\delta\alpha $ In fact, non-empty can only be single path. $a$

10 33 Uniqueness of GSP s in the Half-Plane Take $G_{N}=[-N, N]\cross[0,2N]\cap \mathbb{z}^{2}$ vertical free b.c. and let $Narrow\infty.$ with horizontal periodic and Horizontal but not vertical translation invariance Theorem (Arguin, Damron, Newman, Stein [ADNS]) In the half-plane, $)$ the limit $K=K_{J}v(dJ)$ exists (no subseq.); $\nu K_{J}$ is supported on a single $GSP.$ This is the first complete result for $d>1.$ Strategy show interface between replicas must be empty.

11 34 Interface of GSP s in the Half-Plane $\alpha\delta\alpha \neq\emptyset$ If, then there are infinitely many tethered paths:

12 $\triangleright$ $\triangleright$ $\triangleright$ 35 From Half to Full Plane Let (along subseq.) where is the $\mu^{*}=\lim_{karrow\infty}\frac{1}{k}\sum_{l=1}^{k}t^{-l}\mu$ $T$ vertical shift and $\mu$ is distrib. of. $(\alpha, \alpha, J)$ $\triangleright\mu^{*}$ is a measure on the full plane, translation invariant in full plane by construction. Because we see many $\alpha\triangle\alpha \neq\emptyset$ tethered paths, if, then $\alpha\triangle\alpha $ is not a single path, contradicting full plane result of [NS2]. To do list for the future: Uniqueness of GSP s in the full plane. Uniqueness (or not) of GSP s for $d>2$? Metastates applied to $\beta<\infty.$

13 36 References [ADNS] Arguin, L.- $P$., Damron, M., Newman, C. $M$., Stein, D. $L$.: Commun. Math. Phys. 300, (2010) [AW] Aizenman, M., Wehr, J.: Commun. Math. Phys. 130, (1990) [EA] Edwards, S., Anderson, P. $W$.: (1975) J. Phys. $F5, $ [M] Middleton, A. $A$.: Phys. Rev. Lett. 83, (1999) [NSl] Newman, C. $M$., Stein, D. $L$.: Phys. Rev. Lett. 76, (1996) [NS2] Newman, C. $M$., Stein, D. $L$.: Commun. Math. Phys. 224, (2001) [PY] Palassini, $M$, Young, A. $P$.: Phys. Rev. $B60,$ $R9919-R9922$ (1999) [SK] Sherrington, $D$, Kirkpatrick, S.: Phys. Rev. Lett. 35, (1975)

数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (2011), B25:

Non-existence of certain Galois rep Titletame inertia weight : A resume (Alg Related Topics 2009) Author(s) OZEKI, Yoshiyasu Citation 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (2011), B25: 89-92 Issue Date 2011-04